The Weak Borel conjecture for closed aspherical manifolds

Let $M$ and $N$ be closed aspherical manifolds with isomorphic fundamental groups, and let $\widetilde{M}$ and $\widetilde{N}$ denote their universal covers. Weak Borel conjecture. The universal covers $\widetilde{M}$ and $\widetilde{N}$ are homeomorphic. The conjecture is a proposed stability statement for universal covers of closed aspherical manifolds; the source does not provide evidence of resolution.