Lück's homological torsion growth conjecture for aspherical manifolds

Let $M$ be an aspherical closed manifold. Consider a descending chain of subgroups

$$\pi_1(M)=G_0\supseteq G_1\supseteq G_2\supseteq\cdots$$

such that each $G_i$ is normal in $G=\pi_1(M)$, the index $[G:G_i]$ is finite, and $\bigcap_{i\geq 0}G_i=\{1\}$. Let $p\colon\overline{M}\to M$ be the universal covering and put $M[i]:=G_i\backslash\overline{M}$. Thus $p[i]\colon M[i]\to M$ is a $[G:G_i]$-sheeted covering. Homological torsion growth conjecture. For every natural number $n$ with $2n+1\neq\dim(M)$,

$$\lim_{i\to\infty}\frac{\ln\bigl(\lvert\operatorname{tors}(H_n(M[i];\mathbb{Z}))\rvert\bigr)}{[G:G_i]}=0.$$

If $\dim(M)=2m+1$ is odd, then $\widetilde{M}$ is $\det$-$L^2$-acyclic and

$$\lim_{i\to\infty}\frac{\ln\bigl(\lvert\operatorname{tors}(H_m(M[i];\mathbb{Z}))\rvert\bigr)}{[G:G_i]}=(-1)^m\rho^{(2)}(\widetilde{M}).$$

This conjecture is a homological approximation statement relating normalized torsion growth in finite covers to $L^2$-torsion; for locally symmetric spaces it reduces to a conjecture of Bergeron and Venkatesh. Its general status is unresolved.