The quantum A-polynomial recursion conjecture for the knot-complement series

Let $K\subset S^3$ be a knot, let $F_K(x,q)$ be its knot-complement invariant, and define the normalized series

$$f_K(x,q):=\frac{F_K(x,q)}{x^{1/2}-x^{-1/2}}.$$

Let $\hat{x}$ and $\hat{y}$ be the quantum operators entering the quantum A-polynomial $\hat{A}_K(\hat{x},\hat{y},q)$. Quantum A-polynomial recursion conjecture. For any knot $K\subset S^3$,

$$\hat{A}_K(\hat{x},\hat{y},q)f_K(x,q)=0.$$

This predicts that the normalized knot-complement series is annihilated by the quantum A-polynomial, linking the series to the quantum character variety of the knot complement. The general recursion relation remains open.