Complexification of Kashaev's conjecture

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

$$\lim_{N\to\infty}\frac{\log J_N\left(H;e^{2\pi\sqrt{-1}/N}\right)}{N}=\frac{\operatorname{V}(S^3\setminus H)+\sqrt{-1}\operatorname{CS}(S^3\setminus H)}{2\pi}.$$

This drops the absolute value from Kashaev's conjecture and incorporates the Chern--Simons invariant; the source gives no evidence that the conjecture has been resolved.