The parameterized Volume Conjecture for colored Jones polynomials

Let $K$ be a knot, let $J_N(K;q)$ be its colored Jones polynomial, let $S(u)$ be the classical action appearing in the asymptotic expansion, and let $\mathcal{O}_K$ be an open subset of $\mathbb{C}$.

Parameterized Volume Conjecture. There exists an open subset $\mathcal{O}_K\subset\mathbb{C}$ such that for every $u\in\mathcal{O}_K$,

$$\lim_{N\to\infty}\frac{\log J_N\left(K;\exp\left((u+2\pi\sqrt{-1})/N\right)\right)}{N}=\frac{2\pi}{u+2\pi\sqrt{-1}}S(u).$$

This generalizes the original Volume Conjecture by introducing a parameter on the character variety of the knot complement. The supplied text does not state that the conjecture has been proved in the stated generality.