Greedy strategy gives at most (not optimal but close)
54 days
where the greedy strategy was to
Every day, consider only available permutations that share their triplets with the largest number of unavailable permutations. That is, every day decrease the set of available permutations by the smallest amount possible.
In other words, in terms of graph theory:
I constructed a graph $G$ whose vertices are the permutations, $|V|=8!=40320$.
Two vertices $v,w\in V$ are connected by an edge if and only if they cannot form a solution together. Then, the degree of every vertex will be $d(v)=3623$.
Let $N(v)$ denote the set of neighbors of $v\in V$ and $\overline{N}(v)=N(v)\cup\{v\}$. If we include vertex $v\in V$ in our solution, then vertices in $\overline{N}(v)$ are unusable (cannot be included in our solution).
My greedy strategy was as follows:
Let $X_n\subset V$ and WLOG start with any vertex $v\in V$, i.e. $X_1=\{v\}$.
Let $Y=\bigcup\limits_{x\in X_n} \overline{N}(x)$ and $X_{n+1}=X_{n}\cup\{z\}$ s.t. $z\in V\setminus Y$ maximizes $|N(z)\cap Y|$.
That is, at every step we pick a new vertex $z$ such that the set of unusable vertices $Y$ grows by the smallest amount possible. If there are multiple choices, we can try all of them.
There are 16 scenarios that result in solutions of 54 days, while other scenarios result in less. That is, we cannot get more than 54 days using this greedy strategy.
solution 54 ['01234567', '05634127', '73405126', '71205346', '03476125', '05763124', '13427605', '26305471', '46371052', '16324705', '24063571', '27160354', '52460713', '71560243', '36015724', '26045731', '67145230', '61527304', '27031546', '07624153', '46721503', '24536701', '31425706', '25370461', '36170254', '53014627', '02746531', '06147532', '54261307', '25610473', '75214306', '03647251', '51407236', '40327516', '45103726', '35172064', '31065742', '42107365', '54102673', '74013526', '74165203', '65132074', '41362507', '73562041', '41732650', '07543162', '21376540', '17564023', '50623174', '23745016', '01764235', '67504321', '02164375', '16743502']
solution 54 ['01234567', '05634127', '73405126', '71205346', '03476125', '05763124', '13427605', '26305471', '46371052', '16324705', '24063571', '27160354', '52460713', '71560243', '36015724', '26045731', '67145230', '61527304', '27031546', '07624153', '46721503', '24536701', '31425706', '25370461', '36170254', '53014627', '02746531', '06147532', '54261307', '25610473', '75214306', '03647251', '51407236', '40327516', '45103726', '35172064', '31065742', '42107365', '54102673', '74013526', '74165203', '65132074', '41362507', '73562041', '41732650', '07543162', '21376540', '17564023', '50623174', '23745016', '01764235', '67504321', '16743502', '02164375']
solution 54 ['01234567', '05634127', '73405126', '71205346', '03476125', '05763124', '13427605', '26305471', '46371052', '16324705', '24063571', '27160354', '52460713', '71560243', '36015724', '26045731', '67145230', '61527304', '27031546', '07624153', '46721503', '24536701', '31425706', '25370461', '36170254', '53014627', '02746531', '06147532', '54261307', '25610473', '75214306', '03647251', '51407236', '40327516', '45103726', '35172064', '31065742', '42107365', '54102673', '74013526', '74165203', '65132074', '41362507', '04173265', '73564201', '75620431', '06754321', '21643750', '50162374', '74350621', '31674502', '02137645', '17650423', '65402317']
solution 54 ['01234567', '05634127', '73405126', '71205346', '03476125', '05763124', '13427605', '26305471', '46371052', '16324705', '24063571', '27160354', '52460713', '71560243', '36015724', '26045731', '67145230', '61527304', '27031546', '07624153', '46721503', '24536701', '31425706', '25370461', '36170254', '53014627', '02746531', '06147532', '54261307', '25610473', '75214306', '03647251', '51407236', '40327516', '45103726', '35172064', '31065742', '42107365', '54102673', '74013526', '74165203', '65132074', '41362507', '04173265', '73564201', '75620431', '06754321', '21643750', '50162374', '74350621', '31674502', '02137645', '17650423', '64023175']
solution 54 ['01234567', '05634127', '15627340', '13427560', '25610347', '35641027', '27105346', '60327415', '52760413', '27035416', '60241735', '32471605', '46071352', '43571260', '35026714', '64023715', '74523601', '37405261', '13264705', '42635107', '47265013', '67014235', '17025364', '61370425', '32570614', '05136742', '64213057', '54203617', '07461532', '40376125', '17436520', '62503147', '14507362', '16205473', '15047623', '31750624', '53021764', '21407653', '40157632', '15246730', '47521630', '30754621', '07251463', '63751240', '72154063', '51643207', '73210465', '45731206', '30645172', '16723045', '24531067', '65437201', '20431657', '57243016']
solution 54 ['01234567', '05634127', '15627340', '13427560', '25610347', '35641027', '27105346', '60327415', '52760413', '27035416', '60241735', '32471605', '46071352', '43571260', '35026714', '64023715', '74523601', '37405261', '13264705', '42635107', '47265013', '67014235', '17025364', '61370425', '32570614', '05136742', '64213057', '54203617', '07461532', '40376125', '17436520', '62503147', '14507362', '16205473', '15047623', '31750624', '53021764', '21407653', '40157632', '15246730', '47521630', '30754621', '07251463', '63751240', '72154063', '51643207', '73210465', '45731206', '30645172', '16723045', '24531067', '65437201', '57243016', '20431657']
solution 54 ['01234567', '05634127', '15627340', '13427560', '25610347', '35641027', '27105346', '60327415', '52760413', '27035416', '60241735', '32471605', '46071352', '43571260', '35026714', '64023715', '74523601', '37405261', '13264705', '42635107', '47265013', '67014235', '17025364', '61370425', '32570614', '05136742', '64213057', '54203617', '07461532', '40376125', '17436520', '62503147', '14507362', '16205473', '15047623', '31750624', '53021764', '21407653', '40157632', '15246730', '47521630', '30754621', '07251463', '21540637', '67531240', '64375120', '65432017', '72043165', '30451672', '65172430', '57230164', '23104657', '45372106', '06457321']
solution 54 ['01234567', '05634127', '15627340', '13427560', '25610347', '35641027', '27105346', '60327415', '52760413', '27035416', '60241735', '32471605', '46071352', '43571260', '35026714', '64023715', '74523601', '37405261', '13264705', '42635107', '47265013', '67014235', '17025364', '61370425', '32570614', '05136742', '64213057', '54203617', '07461532', '40376125', '17436520', '62503147', '14507362', '16205473', '15047623', '31750624', '53021764', '21407653', '40157632', '15246730', '47521630', '30754621', '07251463', '21540637', '67531240', '64375120', '65432017', '72043165', '30451672', '65172430', '57230164', '23104657', '45372106', '20645731']
solution 54 ['01234567', '05634127', '51267340', '53467120', '61250347', '31245067', '67501342', '20367451', '16720453', '67031452', '20645731', '36475201', '42075316', '43175620', '31062754', '24063751', '74163205', '37401625', '53624701', '46231507', '47621053', '27054631', '57061324', '25370461', '36170254', '01532746', '24653017', '14603257', '07425136', '40372561', '57432160', '26103547', '54107326', '52601473', '51047263', '35710264', '13065724', '65407213', '40517236', '51642730', '47165230', '30714265', '07615423', '65140237', '27135640', '24371560', '21436057', '76043521', '30415276', '21576430', '17630524', '63504217', '41376502', '60241735']
solution 54 ['01234567', '05634127', '51267340', '53467120', '61250347', '31245067', '67501342', '20367451', '16720453', '67031452', '20645731', '36475201', '42075316', '43175620', '31062754', '24063751', '74163205', '37401625', '53624701', '46231507', '47621053', '27054631', '57061324', '25370461', '36170254', '01532746', '24653017', '14603257', '07425136', '40372561', '57432160', '26103547', '54107326', '52601473', '51047263', '35710264', '13065724', '65407213', '40517236', '51642730', '47165230', '30714265', '07615423', '65140237', '27135640', '24371560', '21436057', '76043521', '30415276', '21576430', '17630524', '63504217', '41376502', '02417365']
solution 54 ['01234567', '05634127', '51267340', '53467120', '61250347', '31245067', '67501342', '20367451', '16720453', '67031452', '20645731', '36475201', '42075316', '43175620', '31062754', '24063751', '74163205', '37401625', '53624701', '46231507', '47621053', '27054631', '57061324', '25370461', '36170254', '01532746', '24653017', '14603257', '07425136', '40372561', '57432160', '26103547', '54107326', '52601473', '51047263', '35710264', '13065724', '65407213', '40517236', '51642730', '47165230', '30714265', '07615423', '23715640', '76514023', '15243607', '73650421', '41735602', '30241576', '52763041', '64135027', '21437605', '17643052', '60435217']
solution 54 ['01234567', '05634127', '51267340', '53467120', '61250347', '31245067', '67501342', '20367451', '16720453', '67031452', '20645731', '36475201', '42075316', '43175620', '31062754', '24063751', '74163205', '37401625', '53624701', '46231507', '47621053', '27054631', '57061324', '25370461', '36170254', '01532746', '24653017', '14603257', '07425136', '40372561', '57432160', '26103547', '54107326', '52601473', '51047263', '35710264', '13065724', '65407213', '40517236', '51642730', '47165230', '30714265', '07615423', '23715640', '76514023', '15243607', '73650421', '41735602', '30241576', '52763041', '64135027', '21437605', '60435217', '17643052']
solution 54 ['01234567', '05634127', '73401562', '75601342', '03472561', '01723564', '53467201', '62301475', '42375016', '52364701', '64023175', '67520314', '16420753', '75120643', '32051764', '62041735', '27541630', '25167304', '67035142', '07264513', '42765103', '64132705', '35461702', '61370425', '32570614', '13054267', '06742135', '02547136', '14625307', '61250473', '71654302', '03247615', '15407632', '40367152', '41503762', '31576024', '35021746', '46507321', '14506273', '74053162', '74521603', '21536074', '45326107', '73126045', '45736210', '07143526', '65372140', '57124063', '10263574', '63741052', '05724631', '27104365', '52743106', '06524371']
solution 54 ['01234567', '05634127', '73401562', '75601342', '03472561', '01723564', '53467201', '62301475', '42375016', '52364701', '64023175', '67520314', '16420753', '75120643', '32051764', '62041735', '27541630', '25167304', '67035142', '07264513', '42765103', '64132705', '35461702', '61370425', '32570614', '13054267', '06742135', '02547136', '14625307', '61250473', '71654302', '03247615', '15407632', '40367152', '41503762', '31576024', '35021746', '46507321', '14506273', '74053162', '74521603', '21536074', '45326107', '73126045', '45736210', '07143526', '65372140', '57124063', '10263574', '63741052', '05724631', '27104365', '06524371', '52743106']
solution 54 ['01234567', '05634127', '73401562', '75601342', '03472561', '01723564', '53467201', '62301475', '42375016', '52364701', '64023175', '67520314', '16420753', '75120643', '32051764', '62041735', '27541630', '25167304', '67035142', '07264513', '42765103', '64132705', '35461702', '61370425', '32570614', '13054267', '06742135', '02547136', '14625307', '61250473', '71654302', '03247615', '15407632', '40367152', '41503762', '31576024', '35021746', '46507321', '14506273', '74053162', '74521603', '21536074', '45326107', '04573621', '73124605', '71260435', '02714365', '65243710', '10526374', '74310265', '35274106', '06537241', '57210463', '21406357']
solution 54 ['01234567', '05634127', '73401562', '75601342', '03472561', '01723564', '53467201', '62301475', '42375016', '52364701', '64023175', '67520314', '16420753', '75120643', '32051764', '62041735', '27541630', '25167304', '67035142', '07264513', '42765103', '64132705', '35461702', '61370425', '32570614', '13054267', '06742135', '02547136', '14625307', '61250473', '71654302', '03247615', '15407632', '40367152', '41503762', '31576024', '35021746', '46507321', '14506273', '74053162', '74521603', '21536074', '45326107', '04573621', '73124605', '71260435', '02714365', '65243710', '10526374', '74310265', '35274106', '06537241', '57210463', '24063571']
(Assuming my python implementation is correct.)