Generalized knots-quivers correspondence

Let $K$ be a knot, and let $F_K$ denote the generating function of its appropriately normalized symmetrically colored HOMFLY-PT polynomials. A symmetric quiver is a quiver whose adjacency matrix is symmetric. Consider the specialization

$$x_i=(-1)^{s_i}a^{a_i}q^{q_i}x^{n_i},\quad i=1,\ldots,m,$$

where the $n_i$ are nonnegative integers. Generalized knots-quivers correspondence. For a given knot, $F_K$ can be written as the quiver generating series of a suitable symmetric quiver with the specialization of variables of the displayed form. This extends the original knots-quivers correspondence by allowing higher powers $x^{n_i}$, which are needed to capture the super-exponential growth of colored HOMFLY-PT polynomials; the statement is presented as the goal of the paper and its general validity remains open.