Exponential torsion growth conjecture for arithmetic hyperbolic manifolds

Let $d=2n+1$ and let $\Gamma\subset \operatorname{SO}_0(d,1)$ be an arithmetic subgroup. Let $\{\Gamma_i\}$ be a cusp-uniform sequence of torsion free congruence subgroups of $\Gamma$ such that $[\Gamma\colon\Gamma_i]\to\infty$ as $i\to\infty$. Put $X_i=\Gamma_i\backslash {\mathbb H}^d$. Let $V_\mathbb{Z}$ be an arithmetic $\Gamma$-module, let $\varrho$ be its associated representation, and let $E_\mathbb{Z}$ be the associated local system of free $\mathbb{Z}$-modules over $X_i$. Exponential torsion growth conjecture. One should have

$$\lim_{i\to\infty}\frac{\log|H^j_{\operatorname{tor}}(\overline{X}_i;E_\mathbb{Z})|}{[\Gamma\colon\Gamma_i]}= \begin{cases} (-1)^n 2t^{(2)}_{\mathbb{H}^d}(\varrho)\operatorname{vol}(\Gamma\backslash\mathbb{H}^d), & j=n+1,\\ 0, & j\neq n+1. \end{cases}$$

This predicts that torsion is exponentially concentrated in the middle degree for congruence towers of arithmetic hyperbolic manifolds, with the rate governed by the corresponding $L^2$-torsion. The preceding results establish lower bounds in important cases, while the general limit and the vanishing assertion in all other degrees remain open.