In and out, and back in again

With White to move, how long will it take for both sides to cooperate, i.e. this is a helpmate, and checkmate Black?

This is based on a problem made made by Otto Blathy. And yes, I know that the postion is illegal.

Currently on mobile, so I'll have to provide the full list of moves later, but I think I have it ending with:

60 Qg4++

The full list of moves:

1. Kxg1 Kg5
2. Kh1 Kg6
3. Kg1 Kg7
4. Kh1 Kf8
5. Kg1 Ke8
6. Kh1 Kd8
7. Kg1 Kc8
8. Kh1 Kb8
9. Kg1 Ka7
10. Kh1 Ka6
11. Kg1 Ka5
12. Kh1 Ka4
13. Kg1 Ka3
14. Kh1 Kb2
15. Kg1 Kc1
16. Kh1 Kd1
17. Kg1 Kxe1
18. Kh1 Kd1
19. Kg1 Kc1
20. Kf1 Kb2
21. Ke1 Ka3
22. Kd1 Ka4
23. Kc1 Ka5
24. Kb1 Ka6
25. Ka2 Ka7
26. Ka3 Kb8
27. Ka4 Kc8
28. Ka5 Kd8
29. Ka6 Ke8
30. Kb7 Kf8
31. Kc8 Kg7
32. Kd8 Kh6
33. Ke8 Kh7
34. Kf7 Kh8
35. Kg6 Kg8
36. Kh6 Kf8
37. Kh7 Ke8
38. Kg8 Kd8
39. Kf8 Kc8
40. Ke8 Kb8
41. Kd8 Ka7
42. Kc8 Ka6
43. Kb8 Ka5
44. Ka7 Ka4
45. Ka6 Ka3
46. Ka5 Kb2
47. Ka4 Kc1
48. Ka3 Kd1
49. Ka2 Ke1
50. Ka1 Kf1
51. Ka2 Kxg2
52. Ka3 Kh3
53. Ka4 g2
54. Ka5 g1=Q
55. Ka6 Qg7
56. Ka7 Qf7
57. exf7 Kh4
58. f8=Q Kh3
59. Qg8 Kh4
60. Qg4#