The pro-fundamental-group Borel conjecture for matchbox manifolds

Let ${\mathcal A}_B$ be a Borel collection of compact manifolds. An ${\mathcal A}_B$-like matchbox manifold is a matchbox manifold modeled on the collection ${\mathcal A}_B$. For a matchbox manifold ${\mathfrak{M}}$, let $\operatorname{pro}\text{-}\pi_1({\mathfrak{M}})$ denote its pro-fundamental-group pro-group. Let ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ be equicontinuous ${\mathcal A}_B$-like matchbox manifolds with isomorphic pro-fundamental-group pro-groups.

Pro-fundamental-group Borel conjecture. If ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are equicontinuous, ${\mathcal A}_B$-like matchbox manifolds that have isomorphic $\operatorname{pro}\text{-}\pi_1$ pro-groups, then ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are homeomorphic.

This is proposed as a stronger version of the preceding shape-based conjecture: an analogue of the implication from isomorphic fundamental groups to homotopy equivalence for aspherical manifolds would reduce it to that conjecture. Its status is not resolved in the supplied text.