The aspherical manifold stability conjecture

Let $M$ be a closed aspherical manifold and let $\widetilde{M}$ be its universal cover. A manifold is inward tame if it has arbitrarily small neighborhoods of infinity that are finitely dominated. The aspherical manifold stability conjecture. The universal cover $\widetilde{M}$ is inward tame. The conjecture is true in dimensions at most $3$ and is weaker than the conjecture that the end of the universal cover is semistable, since inward tameness implies semistability; the general case remains open.