The Volume Conjecture for quantum hyperbolic invariants

Let $(W,L,\rho)$ be a triple consisting of a $3$-manifold, a link, and the relevant representation, with associated quantum hyperbolic invariant $K_N(W,L,\rho)$, refined ${\mathcal I}$-class $\hat{\mathfrak{c}}_{\mathcal I}(W,L,\rho)$, and dilogarithmic invariant

$$R(W,L,\rho):=R(\hat{\mathfrak{c}}_{\mathcal I}(W,L,\rho))\quad \operatorname{mod}\left((\pi^2/2)\mathbb{Z}\right).$$

Real Volume Conjecture. For any $(W,L,\rho)$,

$$\lim_{N\to\infty}\frac{2\pi}{N^2}\log\left(\left|K_N(W,L,\rho)\right|\right)=\operatorname{Im}R(\hat{\mathfrak{c}}_{\mathcal I}(W,L,\rho)).$$

The conjecture is formally compatible with the Volume Conjecture for hyperbolic knots in $S^3$, although the source notes that the relationship between $K_N$ and the coloured Jones polynomial has not been established and requires further investigation.