The Borel conjecture for closed Haken manifolds

Let $M$ and $N$ be closed Haken $n$-manifolds, and let $f:M\to N$ be a homotopy equivalence.

Borel conjecture for Haken manifolds. The map $f$ is homotopic to a homeomorphism.

This is presented as the most fundamental open problem for Haken manifolds. It is a rigidity statement asserting that the homotopy type of a closed Haken manifold determines its homeomorphism type.