The Borel conjecture for shape-equivalent matchbox manifolds

Let ${\mathcal A}_B$ be a Borel collection of compact manifolds of dimension $n \geq 1$. An ${\mathcal A}_B$-like matchbox manifold is a matchbox manifold modeled on the collection ${\mathcal A}_B$. Let ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ be equicontinuous ${\mathcal A}_B$-like matchbox manifolds that are shape equivalent.

Borel conjecture for matchbox manifolds. If ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are equicontinuous, ${\mathcal A}_B$-like matchbox manifolds that are shape equivalent, then ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are homeomorphic.

This conjecture generalizes classification of compact abelian groups by shape to equicontinuous matchbox manifolds. The source notes that the corresponding statement is known for equicontinuous ${\mathbb T}^n$-like matchbox manifolds, but does not resolve the broader ${\mathcal A}_B$ setting.