Stretch factor conjecture for quantum invariants of mapping tori

Let $G=\operatorname{SU}(N)$, let $\Sigma_g$ be a closed surface of genus $g$, and let $\varphi\in\Gamma_g$ be a mapping class with stretch factor $\lambda$. Write $T_{\varphi^n}$ for the mapping torus of $\varphi^n$. Stretch factor conjecture. There exist a rational number $c\in\mathbb{Q}$ and a sequence $\{k_n\}_n\subseteq\mathbb{N}$ such that

$$\lim_{n\to\infty}\sqrt[n]{\left|Z^G_{k_n}(T_{\varphi^n})\right|}=\lambda^c.$$

The conjecture proposes that suitable quantum invariants of iterates of a mapping class detect the exponential dynamics measured by its stretch factor. The preceding $\operatorname{SU}(2)$ calculation motivates the claim, but the supplied text gives no resolution status for the general statement.