What evidence do we have for $\mathsf{UP} \neq \mathsf{NP}$?
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.