The MAF Rigidity Conjecture for approximate fibrations

Let $B$ be a closed aspherical manifold, and let $p\colon M\to B$ and $q\colon N\to B$ be MAFs. MAF Rigidity Conjecture. The maps $p$ and $q$ are controlled homeomorphic if and only if there is a homeomorphism $h\colon M\to N$ such that $q\circ h$ is homotopic to $p$. This is a rigidity statement asserting that the controlled-homeomorphism classification of MAFs is determined by the corresponding ordinary homotopy data. The supplied passage attributes the conjecture to Hughes, Taylor, and Williams and gives no indication that it has been resolved.