Problems that are largely believed to be true, but are unresolved

What about $P \neq NP$? Scott Aaronson has made some excellent points at here

The Riemann hypothesis is largely believed to be true, and further conjectures have been made based on its truth (e.g. statements about the distribution of prime numbers) but no one has ever proved it.

It is widely believed that the fundamental axioms of set theory ZFC are consistent, but this has not been proved in ZFC and in fact provably cannot be proved in ZFC itself unless ZFC is inconsistent, by the second Gödel Incompleteness theorem.

Indeed, whatever fundamental axioms you favor, whether PA or KP or Z or ZF or ZFC or ZFC+large cardinals, then it is natural to suppose also that since you believe that those axioms are true that you also believe that those axioms are consistent, but this is provably not provable from your axioms, unless they are inconsistent.