Shchepin's local triviality conjecture for Serre fibrations

Let $p\colon E\to B$ be a Serre fibration, where $B$ is a locally arcwise connected metric space, and suppose that every fiber of $p$ is homeomorphic to a fixed manifold $M^n$ of dimension $n\leq 4$. Shchepin's conjecture. The fibration $p$ is locally trivial. This is a central problem concerning when a Serre fibration with manifold fibers is a locally trivial bundle. The paper proves existence of local sections for fibrations with fibers homeomorphic to a fixed compact three-dimensional manifold, but the stated local-triviality claim remains unresolved in the supplied context.