As a commenter pointed out, I have not addressed Black's moves other than 69...hxg5 in response to the proposed 69.g5. I attempt to fill that gap here. As a reminder, here is the position we are looking at:
There are two sorts of alternatives to 69...hxg5 to consider here: 69...h5 and various waiting moves with the Black king. Before I begin, though, I should note that I focused on 69...hxg5 because it seemed the most direct way for Black to achieve his goal: eliminating the remaining White pawns. As we will see, however, this is not the only way that Black can achieve a draw in this ending.
It seems foolish at first sight to leave White with the advanced pawn on g5, but Black has a very specific goal in mind. If Black can capture the pawn on g2, force White to advance the other pawn to g6, and get his King back in time, then he achieve the following fortress position:
White cannot force Black's king away from e7 and f8: if White's king gets too close, then it is stalemate.
Unfortunately for Black, however, White seems to be able to avoid this result. The simplest way is to attack the g7-pawn with the King:
Now Black can either attack the pawn on g2 or start pushing his h-pawn, but neither achieves the draw:
a1) 70...Ke2 71.Kf5 Kf2 72.Bd5 a2 73.Bxa2 Kxg2 74.Kg6 h4 75.Be6
The bishop will capture Black's h-pawn, the White king will capture on g7, and the White pawn will promote.
a2) 70...h4. The trick here is that in some lines Black's king is going to attack the g5-pawn instead of the g2-pawn. White has to be careful.
71.Kf5 Ke3 72.Kg6 Kf4
Now if White waits with, say, 73.Ba2, there is probably no win: 73... Kg3 74.Kxg7 Kxg2 75.Bd5+ (or 75.Kh6 h3 76.g6 h2 77.Bd5+ Kf2 78.g7 a2 79.g8=Q h1=Q+ 80.Bxh1 a1=Q. We now reach an endgame of queen and bishop versus queen, which is very complicated, but where the defending side can usually draw.) 75...Kf2 76.Kh6 h3 77.g6 h2 78.g7 a2 79.g8=Q h1=Q+ 80.Bxh1 a1=Q, again reaching the drawish Queen and Bishop versus Queen. In fact, this position is a tablebase draw.
To avoid this outcome, White needs to play:
73...Kg3 (or 73...Ke5 74.Ba2 Kf5 75.Kxh4 Kf4 [75...g6 76.g3 Ke5 77.Kg4 Ke4 78.Bb3 Ke5 79.Kf3 Kf5 80.g4+ Kxg5 81.Kg3 wins] 76.Kh5 Kf5 [76...Kg3 77.Kg6 Kf4 78.g3+ Kg4 79.Be6+ Kxg3 80.Kxg7 wins] 77.g3! g6+ [77...Ke5 78.Kg6 wins] 78.Kh6 Kg4 79.Kxg6 wins.]
74.Bd5 h3 (74...a2 75.Bxa2 Kxg2 76.Kg6 h3 77.Kxg7 h2 78.Bd5+ wins) 75.gxh3 Kxh3 76.g6! (The only winning move).
76...Kg3 77.Kg5 Kf2 78.Kf5 Ke3 79.Ke6 Kd4 80.Ba2 and White wins.
Better yet, and much simpler than these lines, White could diverge earlier with a waiting move like 72.Bf7!
This prevents the Black king from reaching f4 and after 72...Kf2 73.Bd5 Kg3 (73...a2 74.Bxa2 Kxg2 75.Kg6 h3 76.Kxg7 h2 77.Bd5+ wins) 74.g6! (or 74.Ke6!) h3 (74...a2 75.Bxa2 Kxg2 76.Ke6 h3 77.Kf7 h2 78.Bd5+ wins) 75.gxh3 Kxh3 76.Ke6 Kh4 77.Kf7 wins.
Now let's look at what happens if Black plays a waiting move with his king at move 69.
b) 69...Kd4 70.Kf4 Kd3 71.Kf5 Kd4 (...hxg5 or ...h5 at any of these moves will transpose to lines I have already looked at) 72.Ba2 (or another bishop move, but not 72.Kg6? Ke5! 73.Ba2 Kf4 and Black draws) Ke3 73.g6! h5 74.Ke6 h4 75.Kf7 and White wins.
In sum, it appears that White wins in all lines, though it is far from easy to achieve in some of them.
20 hours ago