Torsion Singer conjecture for homology growth

Let $M^n$ be a closed aspherical $n$-manifold with residually finite fundamental group, and set $G=\pi_1(M^n)$. Let $G_k\mathrel{\lhd}G$ be any normal chain of finite-index subgroups satisfying $\bigcap_kG_k=1$. Write $\operatorname{logtor}H_i(G_k;\mathbb Z)$ for the logarithm of the order of the torsion subgroup of $H_i(G_k;\mathbb Z)$. Torsion Singer conjecture. If $i\ne(n-1)/2$, then

$$\limsup_k\frac{\operatorname{logtor}H_i(G_k;\mathbb Z)}{[G:G_k]}=0.$$

This is a torsion-homology-growth analogue of Singer's conjecture for residually finite closed aspherical manifolds. The statement is presented as a motivation for the paper's results, and no resolution is supplied here.