Resolution conjecture for generalized manifolds

Let $X$ be a generalized $n$-manifold, meaning a locally compact $n$-dimensional $ANR$-space that is a $\mathbb Z$-homology $n$-manifold. The space $X$ is resolvable if there is a proper surjective map $f:M\to X$ from a manifold $M$ such that each fiber $f^{-1}(x)$ is contractible in every neighborhood of itself in $M$.

Resolution conjecture. Every generalized $n$-manifold for $n\geq 3$ is resolvable.

The conjecture is true in dimension $3$ only insofar as the source says that it implies the Poincaré conjecture and that partial cases are known. It is false in higher dimensions: non-resolvable generalized $n$-manifolds exist for every $n\geq 6$.