Small covers of the dodecahedron and the $120$-cell

Let $P$ be the right-angled hyperbolic dodecahedron or $120$-cell, and let $W$ be the group generated by reflections across codimension-one faces of $P$. We prove that if $\Gamma\subset W$ is a torsion free subgroup of minimal index, then the corresponding hyperbolic manifold ${\mathbb H}^n/\Gamma$ is determined up to homeomorphism by $\Gamma$ modulo symmetries of $P$.