Quinn's equivariant and stratified rigidity conjecture

Let $M^n$ be a manifold with a “nice” group action or a space with a “nice” stratification, and let $N^n$ be another manifold or space that is equivariantly or stratified homotopy equivalent to $M^n$, respectively. Quinn's conjecture. Then $N^n$ and $M^n$ are equivariantly or stratified homeomorphic, respectively. This loosely stated conjecture extends rigidity from manifolds to spaces with group actions or stratifications; the source gives no resolution status.