The Borel Conjecture for universal covers

Let $M$ and $N$ be closed aspherical manifolds with isomorphic fundamental groups. Their universal covers are contractible open manifolds. The Weak Borel Conjecture. The universal covers of $M$ and $N$ are homeomorphic. This is a weakening of the Borel Conjecture, which asks whether the manifolds themselves are homeomorphic. The source proves the claim for manifolds of dimension $n>4$ whose fundamental groups admit $\mathcal{Z}$-structures, including examples with hyperbolic and CAT(0) fundamental groups, but leaves the general statement open.