The Melvin–Morton–Rozansky conjecture for the knot invariant $F_K$

Let $K\subseteq S^3$ be a knot. Write $F_K(x,q)$ for the knot invariant, set $q={\rm e}^{\hbar}$, and keep $x=e^{n\hbar}$ fixed, where $n$ is the color of $K$. Let $\Delta_K(x)$ be the symmetrized Alexander polynomial, and let $P_r(x)\in\mathbb{Q}[x^{\pm1}]$ with $P_0(x)=1$. Melvin–Morton–Rozansky conjecture. The asymptotic expansion of $F_K(x,q)$ about $\hbar=0$, normalized by $x^{1/2}-x^{-1/2}$, is

$$\frac{F_K\big(x,q={\rm e}^{\hbar}\big)}{x^{1/2}-x^{-1/2}}=\sum_{r=0}^{\infty}\frac{P_r(x)}{\Delta_K(x)^{2r+1}}\hbar^r.$$

This identifies the expansion of $F_K$ with the MMR expansion of the colored Jones polynomial in the large-color limit; the source states that this conjecture was proven in the cited work.