The Borel Conjecture asserts that homotopy equivalent aspherical closed manifolds are homeomorphic, which is still open in general. But, for three-dimensional manifolds, this conjecture holds (I read this in Bessieres-Besson-Boileau), whose proof depends on the geometrization theorem (Perelman).

Question: Does the relative version of the Borel conjecture also hold for compact 3-manifolds with boundary (by the geometrization)? The relative version: If there is a homotopy equivalence between two compact aspherical manifolds that is a homeomorphism between their boundaries, are those manifolds homeomorphic?