If we phrase the Mostow-Prasad rigidity theorem algebraically, it goes like this (let $\mathcal{H}^n$ be a model for hyperbolic $n$-space).
For $n>2$: if $\Gamma,\Delta<\mathrm{Isom}(\mathfrak{h}^n)$ are discrete, isomorphic as groups, and $\mathrm{vol}(\mathcal{H}^n/\Gamma),\mathrm{vol}(\mathcal{H}^n/\Delta)<\infty$, then $\Gamma$ and $\Delta$ are conjugate.
The topological interpretation generally given of this in the literature (Ratcliffe, Benedetti-Petronio, Bonahon) is that in dimension greater than $2$, if two finite-volume hyperbolic manifolds are homeomorphic, then they are isometric.
Is there anything stopping us from replacing the word "manifolds" with the word "orbifolds" there? The algebraic statements seems to translate the same way regardless of torsion.