Rank gradient of small covers

We prove that if $M⟶P$ is a small cover of a compact right-angled hyperbolic polyhedron $P$ then $M$ admits a cofinal tower of finite sheeted covers with positive rank gradient. As a corollary, if ${\pi }_1(M)$ is commensurable with the reflection group of $P$, then $M$ admits a cofinal tower of finite sheeted covers with positive rank gradient.