Akutsu–Kashaev invariant conjecture for hyperbolic knots

Let $M$ be a closed oriented compact $3$-manifold and let $K\subset M$ be a hyperbolic knot. A two-parameter family is a family of smooth functions $J_{M,K}^{(\mathrm{b},N)}(x,j)$ on $\mathbb{R}\times\mathbb{Z}/N\mathbb{Z}$, parametrized by $(\mathrm{b},N)$. For any fully balanced shaped ideal triangulation $X$ of the complement of $K$ in $M$, let $\lambda$ be a gauge-invariant real linear combination of dihedral angles and let $\phi$ be a gauge-noninvariant real quadratic polynomial of dihedral angles. For any one-vertex shaped $H$-triangulation $Y$ of $(M,K)$, let $\varphi$ be a real quadratic polynomial of dihedral angles, and let $\tau\colon \Delta_1(Y)\to\mathbb{R}$ take the value $0$ on $K$ and $2\pi$ on all other edges.

Akutsu–Kashaev conjecture. There exists such a family with the following properties:

$$Z_{\mathrm{b}}^{(N)}(X)=e^{i c_{\mathrm{b}}^2\phi}\frac{1}{\sqrt N}\sum_{j=0}^{N-1}\int_{\mathbb{R}}J_{M,K}^{(\mathrm{b},N)}(x,j)e^{i c_{\mathrm{b}}x\lambda}\,\mathrm{d}x.$$

$$\lim_{\omega_Y\to\tau}\mathrm{D}_{\mathrm{b}}\left(c_{\mathrm{b}}\frac{\omega_Y(K)-\pi}{\pi\sqrt N},0\right)Z_{\mathrm{b}}^{(N)}(Y)=e^{i c_{\mathrm{b}}^2\varphi-i\frac{\pi N}{12}}J_{M,K}^{(\mathrm{b},N)}(0,0).$$

$$\lim_{\mathrm{b}\to0}2\pi\mathrm{b}^2N\log\left|J_{M,K}^{(\mathrm{b},N)}(0,0)\right|=-\operatorname{Vol}(M\setminus K).$$

The conjecture proposes a unified description of the level-$N$ Teichmüller TQFT invariant for ideal and $H$-triangulations, with the hyperbolic volume recovered semiclassically. It was originally stated for $N=1$; the supplied text does not establish whether the updated formulation has been proved or disproved.