I think it has to with Rxb7. If Rxb7 Qd8# If Qxb7 Rxb7 Rxb7 Qd8#
However, I just noticed if Rxb7 black can interpose Rc1+. and it could get complicated because of Kh2, then Rxc1 Rxa7 and that should win for white also.
1. Rxb7 and the d7 rook hangs because of the back rank mate threat on d8. Black can try 1. … Qxf2+ to delay things, and on 2. Kh1 or Kh2 play 2. … g6 for some breathing room. (Exchanges on g2 favor White.) E.g., 2. Kh2 g6 3. Rxd7 Rc8 4. Rbb7 d4 5. Rxf7+ Qxf7 6. Rxf7+ Kxf7 7. Qf6+
White wins with 1. Rxb7!!. If black takes the rook with his d7 rook, he is backrank-mated by 2. Qd8#. If he takes with his queen, then after 2. Rxb7, he ends up a full queen down (2… Rc8 3. Rxd7 Kg8 4. Rd8+ Rxd8 5. Qxd8 – or 2… Rc1+ 3. Qxc1 Rxb7 4. Qc8+ Ke7 5. Qxb7+). The other option for him is:
1… Qxf2+ 2. Kh1 Qxg2+ 3. Qxg2 Rxg2 4. Rxd7 g6 (else 5. Rb8#. Note that 4… Ke8 doesn’t work due to 5. Ra7) 5. Kxg2 and white is now two rooks up.
Note that 2. Kh2 Qxg2+ 3. Qxg2 Rxg2+ 4. Kxg2 is only a one-rook advantage.
Rxb7! wins
rxb7 then qxf2 then kh2 then it doesn’t really matter what black does…
1. Rb7 wins for white
1. Rxb7 1-0
Rxb7
1. Rb7 should do it. The rook at d7 is overtaxed in guarding the b7 pawn and the mate from Qd8.
1. Rb7 Rb7
2. Qd8#
So,
1. …..Qf2
2. Kh2 f6 (what else to try?)
3. Rb8 Kf7(Ke7 is mate in one)
4. Qh5 Ke7
5. Qe8#
And,
1. …..Qa4
2. Rb8 loses even more quickly for black.
Also,
1. …..Rcc7
2. Rb8 is mate in 2 or 3 moves.
And, lastly
1. …..Rc1
2. Kh2 Rb1
3. Ra7 wins the queen for a rook since black cannot retake at a7.
I think it has to with Rxb7. If Rxb7 Qd8#
If Qxb7 Rxb7 Rxb7 Qd8#
However, I just noticed if Rxb7 black can interpose Rc1+. and it could get complicated because of Kh2, then Rxc1 Rxa7 and that should win for white also.
I would venture
1.Rxb7
as the Black Rook on d7 has to guard d8 because the White Queen threatens mate.
Black can postpone its fate with
1.. Qxf2+
but after
2.Kh1
there is no viable check left. Black may also try
1.. Rc1+
but after the cool
2.Kh2
The Black Queen falls, and soon her spouse as well.
1. Rb7 looks convincing. Even if 1. Qf2, the King moves and Black has no good moves
Rxb7
Rxc7 and mate in the last line is only avoidable with heavy material loss.
1.Rxb7
and if … Qxf2+ then 2. Kh1 and white wins with the threat of back rank mate.
1. Rxb7 and the d7 rook hangs because of the back rank mate threat on d8. Black can try 1. … Qxf2+ to delay things, and on 2. Kh1 or Kh2 play 2. … g6 for some breathing room. (Exchanges on g2 favor White.) E.g., 2. Kh2 g6 3. Rxd7 Rc8 4. Rbb7 d4 5. Rxf7+ Qxf7 6. Rxf7+ Kxf7 7. Qf6+
White wins with 1. Rxb7!!. If black takes the rook with his d7 rook, he is backrank-mated by 2. Qd8#. If he takes with his queen, then after 2. Rxb7, he ends up a full queen down (2… Rc8 3. Rxd7 Kg8 4. Rd8+ Rxd8 5. Qxd8 – or 2… Rc1+ 3. Qxc1 Rxb7 4. Qc8+ Ke7 5. Qxb7+). The other option for him is:
1… Qxf2+
2. Kh1 Qxg2+
3. Qxg2 Rxg2
4. Rxd7 g6 (else 5. Rb8#. Note that 4… Ke8 doesn’t work due to 5. Ra7)
5. Kxg2 and white is now two rooks up.
Note that 2. Kh2 Qxg2+ 3. Qxg2 Rxg2+ 4. Kxg2 is only a one-rook advantage.