The complexified volume conjecture for hyperbolic links

Let $L$ be a hyperbolic link, let $J_N(L;q)$ be its colored Jones polynomial, let $\operatorname{Vol}(L)$ be the hyperbolic volume of $S^3\setminus L$, and let $\operatorname{CS}^{\mathrm{SO}(3)}(L)$ be the $\mathrm{SO}(3)$ Chern--Simons invariant associated with the Levi-Civita connection. Complexification of Kashaev's conjecture.

$$\lim_{N\to\infty}\frac{\log J_N\left(L;e^{2\pi\sqrt{-1}/N}\right)}{N}=\frac{\operatorname{Vol}(L)+\sqrt{-1}\operatorname{CS}^{\mathrm{SO}(3)}(L)}{2\pi}.$$

This refines the real volume-growth statement by incorporating the Chern--Simons invariant; its real part recovers Kashaev's conjecture. The source gives the claim as a conjectural generalization and does not state a proof in general.