The parametrized volume conjecture for hyperbolic knots

Let $K$ be a hyperbolic knot. For a parameter $u$, let $\rho_u\colon\pi_1(S^3\setminus K)\to\operatorname{SL}(2;\mathbb{C})$ be the irreducible representation specified by the meridian and longitude eigenvalue parameters, let $T_K(\rho_u)$ be its homological adjoint Reidemeister torsion associated with the meridian, and let $S_K(u)$ be the corresponding Chern--Simons potential. Parametrization of the volume conjecture. There exists a neighborhood $U$ of $0$ in $\mathbb{C}$ such that for $u\in U\setminus\pi\sqrt{-1}\mathbb{Q}$,

$$J_N\left(K;e^{(2\pi\sqrt{-1}+u)/N}\right)\underset{N\to\infty}{\sim}\frac{\sqrt{-\pi}}{2\sinh(u/2)}T_K(\rho_u)^{-1/2}\left(\frac{N}{2\pi\sqrt{-1}+u}\right)^{1/2}\exp\left(\frac{S_K(u)}{2\pi\sqrt{-1}+u}N\right).$$

Here $\operatorname{CS}_{u,v(u)}(\rho_u):=S_K(u)-u\pi\sqrt{-1}-uv(u)/4$, and moreover $v(u)=2\frac{d}{d u}S_K(u)-2\pi\sqrt{-1}$. This parametrized asymptotic formula generalizes the preceding volume conjectures; the source presents it as conjectural and does not state a general proof.