Cubulation conjecture for closed hyperbolic manifolds of low dimension or simplest arithmetic type

Let $M = \mathbb{H}^n / \Gamma$ be a closed hyperbolic $n$--manifold. A compact cubical model for $M$ is a compact cube complex $X$ homotopy equivalent to $M$ whose universal cover admits a $\Gamma$-equivariant quasiisometry to $\widetilde M$.

Cubulation conjecture. Assume that either $n \leq 3$ or that $M$ is arithmetic of simplest type. Then, for every $\lambda > 1$, there is a compact cubical model $X$ such that there is a $\Gamma$--equivariant $\lambda$--quasiisometry

$$\widetilde X \longrightarrow \widetilde M.$$

This proposes cubical models with arbitrarily controlled equivariant quasiisometry constants for closed hyperbolic manifolds in the stated classes. The supplied material does not indicate whether the assertion has been proved or refuted.