Topological rigidity of quasitoric manifolds

Quasitoric manifolds are manifolds that admit an action of the torus that is locally the same as the standard action of $T^n$ on $\mathbb{C}^n$. It is known that the quotients of such actions are nice manifolds with corners. We prove that a class of locally standard manifolds, that contains the quasitoric manifolds, is equivariantly rigid, i.e., that any manifold that is $T^n$-homotopy equivalent to a quasitoric manifold is $T^n$-homeomorphic to it.