In the below position, black must win. Is there a way that white can survive for six moves?
I only know some ways that white can survive for five moves as follows.
3b2rk/3P2pp/8/p7/8/2Q5/PP1pppPP/3R3K w - - 0 1 1. h4 exd1=Q+ 2. Kh2 f1=Q 3. Qc4 Qfxc4 4. g4 Qcxg4 5. a3 Qdxg1#
The White pawn must move to give the space for the king, since the queen and rook cannot prevent the black pawns from promoting to queens. Because of the Bishop on e4, white queen must defend h4. She can go b4, c4, d4, h3, f6 or g7. But it seems that nothing works.