The natural-hyperbolic-structure conjecture for prime links

Let $L$ be an $n$-component prime link whose complement has only hyperbolic pieces. Let $\vec M=(M_1,\ldots,M_n)$ be a coloring, with limiting ratios $s_i=\lim_{N\to\infty}M_i/(N+\frac12)$ satisfying $1-\delta<s_i\leq1$ for $i=1,\ldots,n$. Natural-hyperbolic-structure conjecture. There exists $\delta_L>0$ such that

$$\lim_{N\to\infty}\frac{2\pi}{N+\frac12}\log\left|J_{\vec M}\left(L;e^{\frac{2\pi i}{N+\frac12}}\right)\right|=\frac{1}{2\pi}\operatorname{Vol}\left(\mathbb S^3\backslash L;u_i=2\pi i(1-s_i)\right).$$

Here the volume is the sum of the volumes of the hyperbolic pieces equipped with the natural hyperbolic structures determined by the coloring. This conjecture extends the generalized volume conjecture to prime links whose JSJ decompositions contain several hyperbolic pieces; the source gives no resolution status.