Quinn's ANR homology 4-manifold conjecture for Poincaré 4-complexes

Let $X^4$ be a Poincaré $4$--complex, and let $s(X^4)$ denote its total surgery obstruction.

Quinn's conjecture. If $s(X^4)=0$, then $X^4$ is (simple) homotopy equivalent to an ANR homology $4$--manifold.

This conjecture is proposed as a four-dimensional analogue of the surgery-theoretic manifold recognition theorem in dimensions at least five. The supplied source discusses Quinn's strategy for proving it, but gives no resolution.