It’s Black to move. What must Black do in order to help White castle queenside in just 11 moves?
1. ... Kxh2 White has six pawn moves before they have to move their King. Black can then play Bg2-f1-d3-g6 (somewhere in the line, White can play e5) so that White can play hxg6 and promote their pawn.
From there, I believe Black can essentially waste their turns playing Kh1-h2/g1 ad infinitum while White cleans up the left side of the board and eventually plays O-O-O.