The black Q can't leave the diagonal without check because then RxB mate. But if white allows the Black Q to check, then takes the Q, then Black is only left with pxR(Q) and on the next move it's stalemate.
White can remove the checking possibility by 1. Kg7
then if black plays 1... pxR(Q), 2. QxQ and it's mate in 3
so black would have to play 1... either Qd6 or Qe5 (Qc7 loses after 2. BxQ pxR(Q) then 3. RxB, Qe4, 4. RxQ mate).
But after Qd6 or Qe5, then what? NxQ opens an escape for the K on e5. Moving the N at b2 opens an escape at d3.
So how about 2. Bb6. Then either 2...Q (any move along diagonal e5-b8, otherwise RxB mate), 3. d7d8(Q) and if 3...QxQ, 4. RxB mate. If 3...Qd5, 4. QxQ mate.
Does that work?



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