Bonahon–Wong–Yang volume conjecture for the four-puncture sphere

Let $S$ be the four-puncture sphere, let $\varphi$ be a mapping class of $S$, and let $r$ be an irreducible $\varphi$-invariant representation into $\mathrm{SL}_2(\mathbb C)$ with puncture weights chosen as in the setup. For odd $n$, set $q=\exp\left(\frac{2\pi i}{n}\right)$, let $\rho_r$ be the associated irreducible representation of the Kauffman bracket skein algebra, and let $\Lambda_{\varphi,r}^q$ be the determinant-one intertwiner satisfying

$$(\rho_r\circ\varphi_*)(X)=\Lambda_{\varphi,r}^q\circ\rho_r(X)\circ(\Lambda_{\varphi,r}^q)^{-1}$$

for every $X\in\mathcal{K}^q(S)$. Define the mapping torus

$$M_{\varphi}=S\times[0,1]/(x,1)\sim(\varphi(x),0).$$

Bonahon–Wong–Yang volume conjecture. In the above setup,

$$\lim_{n\rightarrow\infty}\frac{1}{n}\log\left|\operatorname{Trace}\Lambda_{\varphi,r}^q\right|=\frac{1}{4\pi}\operatorname{vol}(M_{\varphi}),$$

where $\operatorname{vol}(M_{\varphi})$ is the volume of the complete hyperbolic metric on $M_{\varphi}$. This relates the asymptotic growth of the quantum mapping-class-group intertwiner to the hyperbolic volume of the mapping torus; the conjecture is presented here without evidence of resolution.