Dan, I agree: as far as I can tell, it hinges on the compelled move question with respect to the only non-reversible piece (pawn), and/or whether 0 legal moves is included in the definition of \\(\leq\\). I *think* if pawns are all removed from the board at the start of the game, the question is sidestepped and the reachable positions are strictly preordered, but there might be another corner case I am missing.