The conjecture that closed aspherical Alexandrov 3-spaces are 3-manifolds

An Alexandrov $3$-space is a three-dimensional Alexandrov space; it is aspherical when all its higher homotopy groups vanish, equivalently when its universal cover is contractible. A $3$-manifold is a topological space locally homeomorphic to $\mathbb{R}^3$. Alexandrov-space manifold conjecture. Every closed, aspherical Alexandrov $3$-space is a $3$-manifold. The conjecture is motivated by the theorem in the paper for sufficiently collapsed irreducible spaces and by the stated higher-dimensional result when the universal cover is compact; the unrestricted three-dimensional case is left as a natural conjecture.