The periodic mapping-torus volume conjecture for Kauffman bracket intertwiners

Let $S$ be an oriented surface with negative Euler characteristic, let $\varphi$ be a periodic diffeomorphism, and let $[\gamma]$ be a $\varphi$-invariant smooth character with $\varphi$-invariant puncture weights $p_v$. For $q_n=e^{2\pi i/n}$ with $(q_n)^{1/2}=e^{\pi i/n}$, let $\Lambda^{q_n}_{\varphi,\gamma}$ be the normalized Kauffman bracket intertwiner associated with these data. The periodic mapping-torus volume conjecture. One should have

$$\lim_{n\text{ odd}\rightarrow\infty}\frac{1}{n}\log\left|\operatorname{Trace}\Lambda^{q_n}_{\varphi,\gamma}\right|=0.$$

The mapping torus of a periodic diffeomorphism is a Seifert manifold and has zero simplicial volume, so this predicts that the exponential growth rate of the intertwiner trace vanishes in the non-hyperbolic periodic case. The paper establishes the analogous vanishing result for closed-torus mapping classes, while the general periodic case is presented as a conjecture.