3-dimensional homotopy covering conjecture

Let a P3R group be a finitely presented group whose universal cover is proper homotopy equivalent to a 3-manifold. An end is semi-stable when its fundamental pro-group at infinity has a representative tower with surjective bonding morphisms, and it has pro-(finitely generated free) fundamental pro-group when that pro-group is pro-isomorphic to an inverse system of finitely generated free groups. 3-dimensional homotopy covering conjecture. A finitely presented group is P3R if and only if each of its ends is semi-stable and has pro-(finitely generated free) fundamental pro-group. The paper gives evidence by proving the characterization for qsf groups; the converse in general is presented as open.