Who wins in this endgame with best play?

The endgame in question starts after I blunder my bishop on move 43:

https://www.chess.com/variants/fog-of-war/game/18756771/87/3

Probably black

Before the bishop blunder, black could've forced a draw I think.  Black would alternate between Be4 Ba8 for 50 moves.  And if it gets captured on a8, King safely takes f4 and promotes.

How?

Idk I'm bad at chess so I'm probably wrong but that was my first guess

I would want a draw as black as there is no way of knowing where white has his king as was seen in the game. There is no "Proper" play to solve this problem at this point in the game.

Great question. My analysis below suggests black has a very slight edge of about 51-52%.

Knowledge

On move 44 black knows where all of white's pieces are, but white does not know if black's king is on g7, h7, or h5.

I think that 43...Bd6  pretty strongly implies that 38...Kh5 was played. Black wouldn't be throwing away their bishop like that unless their king was stuck and they were worried about white playing Kg3-Kh3-Kg3 perpetually. While in reality this isn't certain, let's simplify the analysis by assuming that white correctly bets that 38...Kh5 was plays and behaves as if that's definitely true. Let's further assume that black knows white knows this.

This simplifies the analysis in that it allows white to go straight for b3 without fear that they will run into black's king.

First pass analysis

44...Kg4 45. (50% Ke3, 50% Kd4) (50% Kxf4, 50% Kg3)

Here there's a 25% chance that white wins immediately and a 25% chance that black wins immediately and a 50% that 45...Kg3 is played.

Next black will be considering 46...Kxf4 (winning for black if anything other than 45. Ke3 was played, winning for white if 45.Ke3 was played) and 46...Kg4 47...Kxf4 (winning for black if 45. Ke3 was played, and a favoring-white-but-close-to-a-toss-up if 45. Kd4 46. Ke3 was played).

Looking at that last scenario, after 46. Ke3 Kxf7, one line that might be most probable (in which white plays cautiously because white thinks it is their game to lose) is 48. Kxg3 Ke3 49. Kh6 followed by pawn pushes until 52. g8=Q f8=Q 53. Qe8 Kd2 54. Qd6 Qd3. It's super complicated. White is favored, but it's far from guaranteed.

With this line in mind, perhaps the right policy is not to play the earlier moves with exactly 50% probability. Black should slightly favor getting into this scenario since black does have some chance to win here. And similarly, White might therefore have some reason to avoid it. I'm not sure.

So my final guess for now is that black is >50% likely to win with best play, but not by much, since the final scenario after 46. Ke3 Kxf7 only happens ~12.5% of the time and black has slim winning chances in that scenario.

Supposing black wins ~20% of the time  in that scenario, then black's overall chances of winning if each player chooses between the above moves with exactly 50% probability is 52.5%.

My best guess if we ever learn the answer to this one day (which we might if DeepMind or someone makes a Fog of War engine) is less than 52.5%.

My final guess will be that with best play from both sides, black wins 51.5%.

Black might have a larger advantage if white doesn't actually bet that black's king started on Kh3 like I assumed.

UPDATE:

I realized that white can exploit black's decision policy by always playing 45. Ke3 at the beginning rather than 45. Kd4 ever. Doing this, white would win 75% of the time. Of course, if black knows white is going to do this then black can have a guaranteed win by capturing Kxf4 on move 46 when it is undefended.

I'm not sure what the implications of this are. Maybe it's that white will play 45. Ke3 a little more often and that black will play 45...Kg3 a little more often and 46....Kxf4 a little more often. Possibly black will also consider playing 47...Kg4-Kg3 some fraction of the time rather than always playing 47... Kg4xf4.

But wait, if black knows that white is going to play 45.Ke3 more than 50% of the time, then black can always play 45... Kg3 46...Kxf4 and win more than 50% of the time.

Maybe this means there's not a Nash equilibrium solution? I've never studied game theory and am not sure how to figure this out.

Though wait, I already said black has a slight edge, so maybe white is only going to play 45.Kg3 51% of the time or something barely over 50% like that.

Now that I've thought about this for a while I really want to know the optimal solution.