Hex conjecture (maybe it's trivial to prove/disprove)

The conjecture is: among the winning moves the opponent would get if you would skip your turn (if there would be at least one), there is a move that is winning for you (in the current position!).

I can't think of any obvious counter-example. Any idea? :)

If your opponent is already winning, this cannot be true. If you are already winning, it must be true and I think I've seen it proven.

Yes, sorry, i've forgot to rule out that case. About it being already proven, are you thinking to the "mustplay" regions papers? I don't think it is exactly the same thing.

I think this conjecture is false. A counter example in 4x4:

Blue stones (horizontal) at A1, A2, D2, D3

Red stones (vertical) at A4, B4, C4, D4

Red to move.

If red passes, blues only winning moves are at C1 or B3.

However the only winning moves for red is B2 or C2.

yes, that example is enough to disprove.. thank you Jonatan :)