The only mathematical conjectures worth caring about are falsifiable empirical predictions (i.e. Pi_1 arithmetic statements).
>muh twin prime conjecture
What does it say in practical terms? Absolutely nothing. It has zero empirical meaning. There is no possible data you could look at that would confirm or refute it, apart from a proof in either direction. The question thus is too meaningless to deserve discussion. If you still like it too much, you can transform it to an actually meaningful and empirically falsifiable statement of "there is no proof of twin prime conjecture in ZFC", which is a Pi_1 statement.