Bridson–Reid's profinite rigidity conjecture for hyperbolic 3-manifolds

Let $\mathcal{M}$ and $\mathcal{N}$ be hyperbolic $3$-manifolds. Their profinite completions are the inverse limits of their finite quotient groups. The manifolds are commensurable if they share a common finite-degree covering space. Bridson–Reid's profinite rigidity conjecture. If the profinite completions of $\pi_1(\mathcal{M})$ and $\pi_1(\mathcal{N})$ are isomorphic, then $\mathcal{M}$ and $\mathcal{N}$ are commensurable. This is described in the paper as a notorious open problem and is presented as motivation for a weakening related to the congruence subgroup property.