Symmetric-skew mirror symmetry conjecture for colored HOMFLY-PT polynomials

Let $B$ be a colored, oriented braid, and let $P(B)$ be the invariant of braid conjugacy classes with values in

$$P(B)\in \mathbb{Z}[q,q^{-1},\xi_1,\xi_2,\ldots]/\mathcal{I},$$

where $\mathcal{I}$ is the ideal of relations between colored circles in $\textbf{GenSp}$. A specialization of the parameters $\xi_i$ is said to give a specialization of $P(B)$. Symmetric-skew mirror symmetry conjecture. There exists a specialization of $P(B)$ which gives a multiple of the $\mathrm{Sym}^k_q$-colored HOMFLY-PT polynomial. Applying the substitution $q\leftrightarrow q^{-1}$ yields the $\bigwedge^k_q$-colored HOMFLY-PT polynomial. This is equivalent to the symmetric-skew mirror symmetry conjecture of Gukov and Stošić; the conjecture concerns the relationship between symmetric- and skew-colored HOMFLY-PT invariants under inversion of $q$.