What evidence do we have for $\mathsf{UP} \neq \mathsf{NP}$? 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.
Even, Selman, and Yacobi conjectured that there does not exist a disjoint $NP$-pair $(A, B)$ such that all separators of $(A, B)$ are $ \le_T^p $-hard for $NP$. This conjecture implies that $UP \ne NP$.

S. Even, A. Selman, and J. Yacobi. The complexity of promise problems with applications to public-key cryptography. Information and Control, 61:159–173, 1984.