Exponential growth conjecture for mapping-torus Turaev–Viro invariants

Let $\Sigma$ be a compact orientable surface and let $f\in\mathrm{Mod}(\Sigma)$. Let $M_f$ denote the mapping torus of $f$, and let $\lVert M_f\rVert$ denote its Gromov norm. A compact oriented $3$-manifold is q-hyperbolic when $lTV(M)>0$. Exponential growth conjecture. The mapping torus $M_f$ is q-hyperbolic if and only if

$$\lVert M_f\rVert>0.$$

The conjecture proposes that positivity of Turaev–Viro exponential growth is exactly equivalent to positive simplicial volume for mapping tori. It is described as a more robust conjecture than the Chen–Yang volume conjecture and is supported by computations and prior results cited in the source.