Integrality conjecture distinguishing pseudo-Anosov mapping tori

Let $\Sigma$ be a compact oriented surface and let $f\in\mathrm{Mod}(\Sigma)$ be pseudo-Anosov. Let

$$M_f=\Sigma\times[0,1]/_{(x,1)\sim(f(x),0)}$$

be the mapping torus of $f$, and let $TV_r(M_f)$ denote its $SO(3)$ Turaev–Viro invariant at odd level $r$. Integrality conjecture. If $f$ is pseudo-Anosov, then there can be at most finitely many odd integers $r$ such that

$$TV_r(M_f)\in\mathbb{Z}.$$

The conjecture is proposed to distinguish Turaev–Viro invariants of mapping tori of pseudo-Anosov mapping classes from those of periodic mapping classes. The paper proves the analogous finiteness conclusion under the hypothesis $lTV(M_f)>0$, but does not establish that hypothesis for every pseudo-Anosov mapping torus.