Originally posted by Fat Lady
Black has a sneaky Bd7 move, with the idea of blocking the back rank check with Be8 - after Qxe8+ White's queen is on a white square and so cannot continue the checks after Kg7.
E.g.
1. h5 gxh5
2. Kh4 Bd7
3. Qf6?? Qxd2
4. Qd8+ Be8!
5. Qxe8+ Kg7 and Black wins
White needs to find a better reply to 2. ... Bd7 if 2. Kh4 is supposed to be the drawing move.
2. Kh4! is indeed the drawing move. The game continued
2. ..Rb2 (? - loses a tempo)
3. Qf6 Rb8
4. Kxh5! Qxd2
5. Kh6 Kf8
6. Qh8+ Ke7
7. Qxb8 Qxd1
and ended in a draw by perpetual after 8. Qd6+ Ke8 9. Qb8+ Ke7 10. Qd6+ Ke8 11. Qb8+ Ke7 etc.
When the starting position was reached my opponent and I were already in some time trouble so I doubt we would have found all the best answers all the time.
There are a number of better tries for black at move 2: Bf5, Bg4, Rb8, and Bd7 like you said.
However, I do think it will end in a draw if white applies the best defense.
After 2. ..Bg4 or 2. ..Bd7 white can draw by sacrificing another pawn
3. e6!
A pawn sac to open more lines for the white queen!
If black takes the pawn with his f-pawn, white plays 4. Qf6 after which a perpetual is inevitable (If black then would continue Rb8?! white may even have some winning chances after 5. g6!), whether the bishop is on d7 or g4 is irrelevant then.
So black replies 3. ..Bxe6 and the lines transpose.
3. ..Bxe6
4. Qf6! Rb8
5. Qf4!
This move threatens 2 different things: Qxb8+ and the king march to h6. If black moves his rook off the back rank, white simply plays Qf6, repeating moves. So:
1) 5. ..Rf8
This move protects f7 but at the same time lets white simply march his king up to h6.
6. Kxh5! Bf5!
Because Qf6 must be prevented at all costs.
7. Kh6! Ra8
8. Qd6!
Preventing Ra6+ and threatening Qf6. Note that if black would now play Be6?? he would even lose after Qe5! Kf8 Qxc5+! Kg8 (Ke8 Qc6+) Qxd4 when white picks up a few pawns.
8. ..Qxd2
9. Qf6 Kf8
10. Qh8+ Ke7
11. Qf6+!
The simplest. Now Kf8 leads to repetition, Kd7?? even loses after Qxf5+, while Ke8 is answered by Qc6+.
second line:
2) 5. ..Ra8
This move prevents white from moving his king to h6, but loosens the defense of f7, which white will now try to exploit.
6. Bxh5!
It turns out that black can't try anything in this position. Qe1+ runs into Rf2 and Qh1+ into Rh2, his rook is bound to the protection of the back rank and he can't move his bishop because of Qxf7+.
I'll try to work out the other lines (2. ..Rb8 and 2. ..Bf5) soon enough, but the position is very complex so it might take some time.