Torus-link reformulated colored HOMFLY polynomial structure conjecture

Let $T(rl,kl)$ be a torus link, where $r$ and $k$ are relatively prime, and let $n=n_1+\cdots+n_l$. Let $f_{\lambda^1,\dots,\lambda^l}(t,\nu)$ be the reformulated colored HOMFLY polynomials, and let $s_{\lambda;q}(\mathbf{x})$ denote the symmetric functions defined in the source. For partitions $\lambda^i\vdash n_i$, define $g_{\lambda^1,\dots,\lambda^l}^{\lambda}(t)$ by the asserted identity. Torus-link structure conjecture. After the specialization $t^{1/2}=q^{-1}$ and $\nu^{1/2}=q^{-N}$,

$$\begin{aligned} &\sum_{\lambda^1\vdash n_1,\dots,\lambda^l\vdash n_l}f_{\lambda^1,\dots,\lambda^l}(t,\nu)s_{\lambda^1}(\mathbf{x}_1)\cdots s_{\lambda^l}(\mathbf{x}_l)\\ &=\sum_{\lambda^1\vdash n_1,\dots,\lambda^l\vdash n_l}(q^k-q^{-k})^{-2}q^{-k(r-1)nN}\sum_{\lambda\vdash rn}g_{\lambda^1,\dots,\lambda^l}^{\lambda}(q^{2k})\\ &\qquad\cdot s_{\lambda;q^k}(q^{N-1},q^{N-3},\dots,q^{-(N-1)})s_{\lambda^1;q^k}(\mathbf{x}_1)\cdots s_{\lambda^l;q^k}(\mathbf{x}_l), \end{aligned}$$

where $g_{\lambda^1,\dots,\lambda^l}^{\lambda}(t)\in\mathbb{Z}[t^{\pm1}]$ is invariant under $t\mapsto t^{-1}$. This is presented as a new structure suggested by calculations for torus links; the source does not establish it in general.