One of the FIDE rules states that if someone's flag falls, and there exists a legal sequence of moves such that the other player mates the first player, then the position is a win for the second player. This got me thinking whether this rule can theoretically be hard to enforce for arbiters, i.e. whether it is possible that the arbiter cannot (easily) decide whether a game is winnable for one side or not:
Do there exist "hard" chess puzzles where the objective is to find a sequence of moves of any length, with both sides helping, so that one side wins? So essentially helpmate puzzles, but without specifying the number of moves until mate? Or is it always rather straightforward to determine whether, from a given position, there is a sequence of moves leading to mate?
Perhaps one way to make the job for the arbiter difficult is to not keep track of previous moves, and then present a position to the arbiter (when the flag falls) where it is difficult to prove whether one side can legally castle or not, or take en passant or not etc. - if such positions can only be won when say en passant is available, the arbiter (or the second player claiming a win on time, rather than a draw) would have to construct a proof game to show that the game can indeed be won.
In any case: I cannot come up with examples which are hard helpmates of any length, but I am in no way an expert when it comes to helpmates. Any thoughts or comments are appreciated!
Edit: This question is not about what is sufficient mating material, but about whether positions exist for which it is hard to decide if a mating sequence exists. This is more in the territory of artificially constructed helpmates/proof game problems than about simple, realistic adjournment situations for arbiters.