Andersen–Kashaev's conjecture on Teichmüller TQFT knot invariants

Let $M$ be a closed oriented 3-manifold. For any hyperbolic knot $K\thinspace\subset M$, consider fully balanced shaped ideal triangulations $X$ of the complement of $K$ in $M$ and one-vertex shaped $H$-triangulations $Y$ of the pair $(M,K)$. Let $\tau:\Delta_1(Y)\to\mathbb R$ take the value $0$ on $K$ and $2\pi$ on every other edge. Andersen–Kashaev's conjecture. There exists a smooth function $J_{M,K}(\hbar,x)$ on $\mathbb R_{>0}\times\mathbb R$ with the following properties: for every such $X$, there are a gauge-invariant real linear combination of dihedral angles $\lambda$ and a gauge-noninvariant real quadratic polynomial of dihedral angles $\varphi$ such that

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

for every such $Y$, there is a real quadratic polynomial of dihedral angles $\varphi$ such that

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

and

$$\lim_{\hbar\to0}2\pi\hbar\log\lvert J_{M,K}(\hbar,0)\rvert=-\operatorname{vol}(M\backslash K).$$

The conjecture proposes a common analytic object relating the partition functions of ideal and Hamiltonian triangulations and recovering hyperbolic volume through a semiclassical limit. Its status is not established in the supplied material.