Gukov–Manolescu conjecture on the $F_K$ series and quantum $A$-polynomial

For every knot $K$ in $S^3$, there should exist a two-variable series

$$F_K(x,q)=\sum_{m=0}^\infty f_m(q)\,x^{m+\frac12}$$

with $f_m\in\mathbb{Z}[[q]][q^{-1}]$. Let $\Delta_K(x)$ be the Alexander polynomial, let $P_k(x)$ denote the polynomials occurring in the Melvin–Morton–Rozansky expansion of the colored Jones polynomial, and let $\hat{A}_k(\hat{x},\hat{y},q)$ be the quantum $A$-polynomial. The notation $\sim$ means that the two sides are related by Borel resummation.

Gukov–Manolescu conjecture. The series satisfies

$$F_K\left(x,e^{\frac{h}{2}}\right)\sim\frac{1}{\Delta_K(x)}+\sum_{k=1}^\infty\frac{P_k(x)}{\Delta_K^{2k+1}(x)}\frac{h^k}{k!},$$

where the right-hand side is the Melvin–Morton–Rozansky expansion of the colored Jones polynomial, and it is annihilated by the quantum $A$-polynomial:

$$\hat{A}_k(\hat{x},\hat{y},q)F_K(x,q)=0.$$

This conjecture packages the expected resummation relation between $F_K$ and the colored Jones polynomial together with a quantum $A$-polynomial difference equation. The supplied text does not state whether the conjecture has been resolved.