Volume conjecture for relative Turaev–Viro invariants

Let $(M,\mathcal T)$ be an ideally triangulated $3$-manifold with edge set $E$, and let $\{\mathbf b^{(r)}\}$ be a sequence of colorings of $(M,\mathcal T)$. For each $i\in\{1,\dots,|E|\}$, define

$$\theta_i=\left|2\pi-\lim_{r\to\infty}\frac{4\pi b_i^{(r)}}{r}\right|$$

and write $\theta=(\theta_1,\dots,\theta_{|E|})$. Here $M_{E_\theta}$ denotes $M$ equipped with the hyperbolic polyhedral metric on $(M,\mathcal T)$ having cone angles $\theta$.

Volume conjecture. As $r$ ranges over all odd integers and $q=e^{\frac{2\pi i}{r}}$,

$$\lim_{r\to\infty}\frac{2\pi}{r}\log\mathrm{TV}_r(M,E,\mathbf b^{(r)})=\operatorname{Vol}(M_{E_\theta}).$$

This relates the exponential growth of the relative Turaev–Viro invariants to the volume of the hyperbolic polyhedral metric determined by the limiting coloring data. The surrounding discussion explains that relative Turaev–Viro invariants encode geometric information, while rigidity results show that such polyhedral metrics are determined up to isometry by their cone angles. The status of the assertion is not resolved in the supplied source context.