This is perhaps not quite what the question is asking for, but it was mentioned in a comment and seems at least tangentially relevant. There are examples of axioms going beyond ZF that contradict each other and yet are both "taken seriously." These axioms are not exactly conjectures in the usual sense of the word, but sometimes people will advocate for them as "basic axioms" to be adjoined to ZF.

1. The (full) axiom of choice contradicts various other axioms such as the axiom of determinacy and the axiom that "all sets of reals are Lebesgue measurable." Although it is standard mathematical practice to accept the axiom of choice as a basic axiom of mathematics, these other axioms are not necessarily treated as being straightforwardly false, in part because they can be compatible with a weak version of choice.
2. The existence of measurable cardinals (or even of sharps) is incompatible with $V = L$, the axiom of constructibility. Large cardinal axioms are widely "accepted" by set theorists, but $V=L$ has a few advocates.
3. Forcing axioms are incompatible with V = Ultimate L (the former imply $2^{\aleph_0} = \aleph_2$ and the latter implies $2^{\aleph_0} = \aleph_1$). There is a nice Quanta Magazine article that gives a nontechnical overview.