Andersen–Kashaev's TQFT volume conjecture for hyperbolic knots

Let $M$ be an oriented compact closed $3$-manifold, and let $K$ be a hyperbolic knot in $M$. For a fully balanced shaped ideal triangulation $X$ of the complementary space of $K$ in $M$, let $Z_{\hbar}(X)$ denote its partition function; for a one-vertex shaped $H$-triangulation $Y$ of $(M,K)$, let $Z_{\hbar}(Y)$ denote the corresponding partition function, let $\omega_Y$ be its weight function, and let $\Phi_{\rm b}$ be the relevant quantum dilogarithm. Define $\tau:\Delta_1(Y)\to\mathbb{R}$ by assigning $0$ to the edge representing $K$ and $2\pi$ to every other edge. Andersen–Kashaev's conjecture. There exists a smooth function $J_{M,K}(\hbar,x)$ on $\mathbb{R}_{>0}\times\mathbb{R}$ such that: (1) for every such $X$, there are a real linear combination $\lambda$ of gauge-invariant dihedral angles and a real second-order polynomial $\phi$ in not necessarily gauge-invariant dihedral angles with

$$Z_{\hbar}(X)=e^{i\frac{\phi}{\hbar}}\int_{\mathbb{R}}J_{M,K}(\hbar,x)e^{-\frac{x\lambda}{\sqrt{\hbar}}}\,dx;$$

(2) for every such $Y$, there is a real second-order polynomial $\varphi$ in dihedral angles such that

$$\lim_{\omega_Y\to\tau}\Phi_{\rm b}\left(\frac{\pi-\omega_Y(K)}{2\pi i\sqrt{\hbar}}\right)Z_{\hbar}(Y)=e^{i\frac{\varphi}{\hbar}-i\frac{\pi}{12}}J_{M,K}(\hbar,0);$$

and (3)

$$\lim_{\hbar\to0}2\pi\hbar\log\left|J_{M,K}(\hbar,0)\right|=-\operatorname{Vol}(M\backslash K).$$

The conjecture proposes a common function governing the partition functions associated with ideal triangulations and one-vertex $H$-triangulations, while its semiclassical limit recovers the hyperbolic volume of the knot complement. It is stated here under the condition $\mathsf{b}\in\mathbb{R}_{>0}$; its general status is not established in the supplied source.