Torsion-volume growth lower bound for biperiodic alternating links

Let $\mathcal L$ be a hyperbolic biperiodic alternating link, and let $\{X(L_n)\}$ be the associated sequence of finite-volume covers described above. Write $TH_1(X(L_n))$ for the torsion subgroup of $H_1(X(L_n);\mathbb Z)$, and let ${\rm vol}(X(L_n))$ denote the hyperbolic volume. Torsion-volume growth conjecture.

$$\lim_{n\to \infty}\frac{\log|TH_1(X(L_n))|}{{\rm vol}(X(L_n))}\geq\frac{1}{4\pi},$$

with equality if and only if $\{X(L_n)\}$ have embedded totally geodesic surfaces for all $n$.

This conjecture proposes a universal lower bound for homological torsion growth normalized by hyperbolic volume, with equality characterized by the presence of embedded totally geodesic surfaces in every member of the covering sequence. The source gives no resolution status.