HOMFLY-PT analogue of $F_K$

Let $K$ be a knot, let $a$ and $q$ be parameters, and let $\hat{A}_K(\hat{x},\hat{y},a,q)$ be the $a$-deformed quantum $A$-polynomial. HOMFLY-PT analogue conjecture. There exists a function $F_K(x,a,q)$ such that

$$\hat{A}_K(\hat{x},\hat{y},a,q)F_K(x,a,q)=0,$$

for every knot $K$, and

$$F_K(x,q^N,q)=F_K^{{\rm SU}(N),{\rm sym}}(x,q).$$

Moreover, it obeys the Weyl symmetry

$$F_K(x^{-1},a,q)=F_K(a^{-1}q^2x,a,q).$$

The conjecture seeks a single $a$-deformed function interpolating the symmetric $\mathrm{SU}(N)$ specializations and satisfying the quantum $A$-polynomial equation. The source presents it as a future direction and gives no general construction or proof.