Turaev–Viro commensurability conjecture for geometric 3-manifolds

Let $M$ and $N$ be closed irreducible geometric 3-manifolds. Their abelian and $SU(2)$ Turaev–Viro invariants are the collections of invariants associated to abelian and $SU(2)$ theories, respectively. Turaev–Viro commensurability conjecture. If $M$ and $N$ have the same abelian and $SU(2)$ Turaev–Viro invariants, then $M$ and $N$ should be commensurable and, in particular, share the same geometry.

The conjecture is motivated by the preceding results for torus bundles, where these invariants impose strong arithmetic restrictions, but the general statement is left open.