Relative Turaev–Viro volume conjecture with adjoint torsion

Let $N$ be a $3$-manifold with non-empty boundary and let $\mathcal T$ be an ideal triangulation with edge set $E$. Let $\{\mathbf b^{(r)}\}$ be a sequence of colorings of $(N,\mathcal T)$ by $\{0,\dots,r-2\}$, define signs $\mu_k$ according as $b_k^{(r)}$ is eventually above or below $r/2$, and set

$$\theta^{(r)}_k=\mu_k\left(\frac{4\pi b_k^{(r)}}{r}-2\pi\right).$$

Assume that the corresponding hyperbolic polyhedral metrics $N^{(r)}$ exist, with volume $\mathrm{Vol}(N^{(r)})$, edge lengths $l_k^{(r)}$, and that $M$ is obtained by doubling $N$ and removing the doubled edges, with holonomy representation $\rho_{M^{(r)}}$ and adjoint Reidemeister torsion $\mathbb{T}_{(M,\mathbf m)}([\rho_{M^{(r)}}])$. Relative Turaev–Viro volume conjecture. If $\{\theta^{(r)}\}$ converges as $r\to\infty$, then, for all positive odd integers $r$ and $q=e^{2\pi i/r}$,

$$\mathrm{TV}_r(N,E,\mathbf b^{(r)})=C\frac{e^{-\sum_{k=1}^{|E|}\mu_kl_k^{(r)}}}{\sqrt{\mathbb{T}_{(M,\mathbf m)}([\rho_{M^{(r)}}])}}r^{\frac32\chi(N)}e^{\frac{r}{2\pi}\mathrm{Vol}(N^{(r)})}\left(1+O\left(\frac1r\right)\right),$$

where $C$ is independent of the geometric structure on $N$ and $\chi(N)$ is the Euler characteristic of $N$. This is a relative Turaev–Viro analogue of volume conjectures, expressing the asymptotics through hyperbolic volume, edge lengths, Euler characteristic, and adjoint torsion. The source notes that related special cases are proved, but the conjecture itself is not established in general.