Since all relevant mates are less than 50 moves after the latest [capture or pawn-move] which was _not_ after the relevant move, the 50-move rule does not affect anything in this post.
After 52. Be6+ , black has 4 legal moves:
Kd8 and Ke8 both allow Rh8#.
Kc6 allows d7 with an eval above [+15].
Kxb6 (played) is a Lomonosov-tablebase
loss for black, as I explain below.
Black has five legal 53's; they all allow Rxb6.
Regardless of which of them black chooses,
after 54. Rxb6 , Ne2 is black's only check, and if that
then Bxe2 is legal and leaves a 7-piece position that
the Lomonosov tablebases say is a win-for-white in at most 16.
If black's 54 is neither that nor Kxb6, then regardless of what
else it is, Rxb7 is legal and leaves a 7-piece position that the
Lomonosov tablebases say is a win-for-white in at most 25.
Finally, 54. ... Kxb6 (played) leaves a 7-piece position that
the Lomonosov tablebases say is a win-for-white in 33.
After 54. ... Kxb6 (played), white has exactly two
winning moves: Kf2 (played), and g4, which wins faster.
One optimal line is 55. g4! Ne4 56. Kg2 Kc5 57. Kf3 Nf6
58. g5 Ne8 59. Kf4 Kd6 60. Kf5 Ke7 61. Ke5! .
After all but the last ply of that, patzer2's
61. Kg6 still wins, but is 4 moves slower.
That's all about white's unique-best 55, which is g4.
What white _played_ was the only
other winning move, 55. Kf2 .
56. Ke3 (played) was the only winning move.
58. b4 lets black draw.
The winning moves were Kd3 and Kd4.
58. ... Ke5 (played) was the only drawing move.
60. ... Nf6 loses. The drawing moves were Nd6,Ng5,Nf2+.
61. Bc6 (played) was the only winning move.
63. Kc3 lets black draw. Bc6 allows Kd6
repeating the position from 4 ply before that,
and Kc4 was the only other winning move.
63. ... Ne8 (played) was the only drawing move.
Black had plenty of ways to keep the draw on black's
66 and 67, and played one of them both times.
66. ... Kxg5 and 67. ... Nxg5 are other drawing moves,
and 6-piece tablebases confirm that those two each draw.
6-piece tablebases also say that 68. ... Kxg5 (played) draws
and that the draw remained for the short rest of the game.