White moves 6P, 5K, 2B leaving 8 for the rook.
So it can go from h1 to h4/h5 (not sure which but leaning towards h5), then to a4/a5, a8 and f8 before returning. Which is all of his accounted for.
So black has a minimum of three pawn moves (to b4 and a5), the rest to shuttle his pieces out and back again. So max of 18 to do this, or 9 each way. K mi ...[text shortened]... can't take place before the 5th move. K to c3, pawns to e3/b3. a5, b5, Nc6 and Rb8 needed first
Well the king may come through e2-d3-c4-b3-a2, after Whiite has moved the a pawn to a3 and before moving the b pawn. Maybe. I don't have time to look at it more carefully now.