Lê's asymptotic volume conjecture

Let $\Gamma$ be the fundamental group of a connected, orientable, irreducible, compact 3-manifold whose boundary is empty or a collection of tori. For finite-index subgroups $\Gamma_n$ tending to the trivial subgroup, consider the normalized torsion in first homology. Lê's asymptotic volume conjecture.

$$\limsup_{\Gamma_n\to\{1\}}\frac{\log\lvert H_1(\Gamma_n)_{\operatorname{tors}}\rvert}{[\Gamma:\Gamma_n]}=\frac{\operatorname{vol}(\Gamma)}{6\pi}.$$

Lê proved the inequality in the direction $\le$, even without requiring the subgroups to be normal; the equality remains the conjectural part.