The manifold-structure conjecture for aspherical Poincaré complexes

Let $X$ be a finitely dominated aspherical Poincaré complex. The manifold-structure conjecture. The complex $X$ is homotopy equivalent to a closed topological manifold. This is one of the source's optimistic conjectures about realizing aspherical Poincaré complexes by manifolds; the source gives no resolution status.