The w.g.s.c. conjecture for fundamental groups of closed aspherical manifolds

Let $\Gamma$ be the fundamental group of a closed aspherical manifold. A finitely presented infinite group $\Gamma$ is w.g.s.c. when some compact polyhedron with fundamental group $\Gamma$ has a w.g.s.c. universal covering space. w.g.s.c. conjecture for closed aspherical manifolds. The group $\Gamma$ is w.g.s.c. The conjecture extends the group-theoretic perspective on geometric simple connectivity of universal covering spaces; the source presents it as a possible general statement and gives no resolution.