Homogeneity conjecture for generalized manifolds

Let $X$ be a connected generalized $n$-manifold, meaning a locally compact $n$-dimensional $ANR$-space that is a $\mathbb Z$-homology $n$-manifold. Assume that $X$ has the disjoint disks property: arbitrary maps $f,g:\mathbb B^2\to X$ can be approximated arbitrarily closely by maps with disjoint images.

Homogeneity conjecture. Every connected generalized $n$-manifold with the disjoint disks property is homogeneous, for $n\geq 5$.

This conjecture is attributed to Bryant, Ferry, Mio and Weinberger. The source gives no evidence of resolution.