The "double" part of double crux means that the crux you choose has to be a crux for both people, i.e. if it's established either way, that alone has to be sufficient to convince both people of the overall argument being true or false.
If you pick a crux that is a truly provably untenable position, and it turns out to also be a crux for the other person then they should happily change their mind on seeing the proof. If they don't it's not like they're committed to saying you've "won" against their will or anything, at worst it turns out that the crux they agreed on wasn't really a crux for them after all. At best, they discover that their belief rested on an untenable position.
By design it's pretty hard for a malicious actor to play double crux to win without also playing it to discover the truth, so long as they actually choose honest cruxes for themselves (and to some extent, even if they don't).