 Relative boundary rigidity from a hierarchy

A map between compact connected orientable irreducible 3-manifolds with nonempty boundaries, injective on the boundary and on the fundamental group, is homotopic to a homeomorphism relative to the boundary. The boundary determines degree $\pm 1$, and the cover corresponding to the image of the fundamental group consequently has one sheet. Straighten the preimage of an incompressible hierarchy surface, cut both manifolds, and repeat. At the terminal balls, extend the prescribed boundary homeomorphisms by coning. The homotopies and homeomorphisms glue across the straightened surface collars.