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$k = 2$ : completed

$k = 3$

in plain text:

A.BB...
....CCC
DD...C.
DD..E..
....EEE
..FF..E
.FFF...

$k = 2$ : possible for $n\in 2..9$

$k = 3$, so far, solutions have been found when $n = 5..9$

in plain text:

A.BB...
....CCC
DD...C.
DD..E..
....EEE
..FF..E
.FFF...

(3,8)

in plain text:

..AA..C.
..AA.D..
BB...D..
B....D.E
....F.EE
....F.EE
.G.FF...
GGG.....

(3,9)

in plain text:

AAA......
A.A.B....
...BB.D..
..BB..D..
.....DDD.
GG......E
.G.....EE
.....F.EE
...FFF...

$k=4$

(4,4), (4,5), (4,6) are impossible, because there is not enough holes to separate the pieces.

(4,7)

in plain text:

AA.B.D.
A.C..DD
A.CCC..
A..C.EE
.GG.F.E
.GG.F.E
.G.FFF.

(4,8)

in plain text:

.AA..BB.
AA....BB
.A..CC.B
D..E.CC.
D.EEE...
DD.E.F..
..G.G.HH
..GGG..H

Perfect boards

$k = 2$

Combined with the (2, 2), (2, 3), (2, 4) and (2, 5) in the first post, this is complete, as $n \geq 10$ requires $k \cdot n \geq 20$ blocks, but if $k = 2$ only 19 blocks can be places (see first post).

(2,6)

in plain text:

.ww...
ww....
w..v..
..vv..
....bb
....bb

(2,7)

in plain text:

A...B..
...BB..
..BB...
.....CC
.....CC
.DD....
DD.....

(2,8)

in plain text:

AA......
AA......
..BB....
..B....C
......CC
.....CC.
...DD...
....DD..

(2,9)

in plain text:

.AA......
.....BB..
.....BB..
CC.......
C......D.
.......DD
....E...D
...EE....
..EE.....

$k = 3$

(3,6)

in plain text:

www...
w..zz.
w...zz
.uuu..
.u.u.v
..p.vv

And for the style : diagonals also have a 3 out of 6 completion.

(3,7)

in plain text:

A.BB...
....CCC
DD...C.
DD..E..
....EEE
..FF..E
.FFF...

Perfect boards

$k = 2$ : completed

Combined with the (2, 2), (2, 3), (2, 4) and (2, 5) in the first post, this is complete, as $n \geq 10$ requires $k \cdot n \geq 20$ blocks, but if $k = 2$ only 19 blocks can be placed (as stated in the question).

(2,6)

plain text version:

.AA...
AA....
A..B..
..BB..
....CC
....CC

(2,7)

in plain text:

A...B..
...BB..
..BB...
.....CC
.....CC
.DD....
DD.....

(2,8)

in plain text:

AA......
AA......
..BB....
..B....C
......CC
.....CC.
...DD...
....DD..

(2,9)

in plain text:

.AA......
.....BB..
.....BB..
CC.......
C......D.
.......DD
....E...D
...EE....
..EE.....

$k = 3$

(3,6)

in plain text:

AAA...
A..BB.
A...BB
.CCC..
.C.C.E
..D.EE

And for the style : diagonals also have a 3 out of 6 completion.

(3,7)

in plain text:

A.BB...
....CCC
DD...C.
DD..E..
....EEE
..FF..E
.FFF...

Perfect boards

(2,6)

in plain text  :

.ww...
ww....
w..v..
..vv..
....bb
....bb

(3,6)

in plain text  :

www...
w..zz.
w...zz
.uuu..
.u.u.v
..p.vv

And for the style : diagonals also have a 3 out of 6 completion.

Perfect boards

$k = 2$

Combined with the (2, 2), (2, 3), (2, 4) and (2, 5) in the first post, this is complete, as $n \geq 10$ requires $k \cdot n \geq 20$ blocks, but if $k = 2$ only 19 blocks can be places (see first post).

(2,6)

in plain text:

.ww...
ww....
w..v..
..vv..
....bb
....bb

(2,7)

in plain text:

A...B..
...BB..
..BB...
.....CC
.....CC
.DD....
DD.....

(2,8)

in plain text:

AA......
AA......
..BB....
..B....C
......CC
.....CC.
...DD...
....DD..

(2,9)

in plain text:

.AA......
.....BB..
.....BB..
CC.......
C......D.
.......DD
....E...D
...EE....
..EE.....

$k = 3$

(3,6)

in plain text:

www...
w..zz.
w...zz
.uuu..
.u.u.v
..p.vv

And for the style : diagonals also have a 3 out of 6 completion.

(3,7)

in plain text:

A.BB...
....CCC
DD...C.
DD..E..
....EEE
..FF..E
.FFF...