Relative Reshetikhin–Turaev volume conjecture with adjoint torsion

Let $M$ be a closed oriented $3$-manifold and let $L$ be a framed hyperbolic link in $M$ with $|L|$ components. Let \\{\mathbf a^{(r)}\} be a sequence of colorings of the components of $L$ by $obreak\{0,\dots,r-2\}$, and define signs $\mu_k$ and cone angles

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

Assume that the corresponding hyperbolic cone metrics $M^{(r)}$ exist for all sufficiently large $r$, with volume $\mathrm{Vol}(M^{(r)})$, Chern–Simons invariant $\mathrm{CS}(M^{(r)})$, logarithmic holonomies $\mathbf u^{(r)}_{\gamma_k}$, holonomy representation $\rho_{M^{(r)}}$, and adjoint Reidemeister torsion $\mathbb{T}_{(M{\smallsetminus} L,\mathbf m)}([\rho_{M^{(r)}}])$. Relative Reshetikhin–Turaev volume conjecture. If $\{\theta^{(r)}\}$ converges as $r\to\infty$, then, for all positive odd integers $r$ and $q=e^{2\pi i/r}$,

$$\mathrm{RT}_r(M,L,\mathbf a^{(r)})=c\frac{e^{\frac12\sum_{k=1}^{|L|}\mu_k\mathbf u^{(r)}_{\gamma_k}}}{\sqrt{\mathbb{T}_{(M{\smallsetminus} L,\mathbf m)}([\rho_{M^{(r)}}])}}e^{\frac{r}{4\pi}(\mathrm{Vol}(M^{(r)})+i\mathrm{CS}(M^{(r)}))}\left(1+O\left(\frac1r\right)\right),$$

where $c$ has norm $1$ and is independent of the geometric structure on $M$. This predicts an asymptotic expansion of relative Reshetikhin–Turaev invariants in terms of geometric data and adjoint torsion; it generalizes volume-conjecture phenomena. The source states that it was proved for special families, while the full assertion is not resolved here.