Following Josh Grochow's suggestion, I am converting my comment from a previous question into a new question.
What evidence do we have for $\mathsf{UP} \neq \mathsf{NP}$?
Here $\mathsf{UP}$ is the class of languages recognizable by polynomial time non-deterministic Turing machines that have a unique accepting path on "yes" instances and no accepting path on "no" instances.
Obviously $\mathsf{UP} \subseteq \mathsf{NP}$, but why would we believe that the containment is strict? The evidence I can find is oracle separation: subject to a random oracle, $\mathsf{P} \subsetneq \mathsf{UP} \subsetneq \mathsf{NP}$. Also, the Complexity Zoo suggests that $\mathsf{UP}$ is not believed to have complete problems.