TITLE: Topological rigidity of compact manifolds in dimension three 
QUESTION [9 upvotes]: 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?

REPLY [12 votes]: Yes.
When the manifolds are Haken this is a theorem of Waldhausen. See Ian Agol's answer here.
Since your manifolds are aspherical, they are irreducible by the Poincaré conjecture.  Since they have boundary and are irreducible, they are Haken.