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](http://www-fourier.ujf-grenoble.fr/~lbessier/english_principal.pdf)), 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?



Topological rigidity of compact manifolds in dimension three 
Yes.


When the manifolds are Haken this is a theorem of Waldhausen. See Ian Agol's answer [here](https://mathoverflow.net/questions/35680/complete-knot-invariant/35687#35687).


Since your manifolds are aspherical, they are irreducible by the Poincaré conjecture. Since they have boundary and are irreducible, they are Haken.


