The volume conjecture for the Teichmüller TQFT

Let $M$ be a closed oriented compact $3$-manifold, and let $K\subset M$ be a hyperbolic knot. For a 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-non-invariant real quadratic polynomial of dihedral angles. Write $\mathop{Z_\hbar}(X)$ for the Teichmüller TQFT invariant of $X$.

The volume conjecture for the Teichmüller TQFT. There exists a smooth function $J_{M,K}(\hbar,x)$ on $\mathbb{R}_{>0}\times\mathbb{R}$ such that

$$\mathop{Z_\hbar}(X)=e^{\mathsf i\frac{\phi}{\hbar}}\int_{\mathbb{R}}J_{M,K}(\hbar,x)e^{-\frac{x\lambda}{\sqrt{\hbar}}}\operatorname{d}\!x,$$

and

$$\lim_{\hbar\to0}2\pi\hbar\log\vert J_{M,K}(\hbar,0)\vert=-\mathop{\fam0 Vol}\nolimits(M\setminus K).$$

This conjecture proposes an asymptotic relation between the Teichmüller TQFT invariant and the hyperbolic volume of a knot complement as $\hbar\to0$. The function and the stated properties are conjectural for every closed oriented compact $3$-manifold and hyperbolic knot satisfying the hypotheses.