Let $p:E\to B$ be a locally trivial fibration of connected, non-compact smooth manifolds. Let $U\subset E$ be a connected open subset and $p|_U:U\to p(U)$ has diffeomorphic fibers.

Can we conclude that $p|_U$ is again a locally trivial fibration?

This question is related to my other question I posted before, but I find this general version interesting in its own right.