Detcherry–Kalfagianni exponential growth conjecture

Let $M$ be a compact, oriented $3$-manifold with empty or toroidal boundary, and let

$$\textit{lTV}(M):=\liminf_{r\rightarrow\infty,\,r\text{ odd}}\frac{2\pi}{r}\log\left|TV_r\left(M;q=e^{\frac{2\pi i}{r}}\right)\right|.$$

The manifold $M$ is $q$-hyperbolic when $\textit{lTV}(M)>0$, and $\lVert M\rVert$ denotes its Gromov norm. Exponential Growth Conjecture. The manifold $M$ is $q$-hyperbolic if and only if

$$\lVert M\rVert>0.$$

This weaker conjecture relates exponential growth of Turaev–Viro invariants to positive Gromov norm; the source does not specify whether it has been resolved.