The Cohomological Rigidity Problem in toric topology asks if two quasitoric manifolds are homeomorphic or diffeomorphic when their integral cohomology rings are isomorphic. As cohomology rings are homotopy invariant, it is natural to ask whether these manifolds are homotopy equivalent under the same conditions, applicable to spaces like toric orbifolds. No counterexamples have been found, and positive results exist in various cases. In this talk, we focus on quasitoric manifolds over products of simplices, showing that they are homotopy equivalent after localization if their integral cohomology rings are isomorphic. This is joint work with T. So, J. Song and S. Theriault.