Symmetric-group action conjecture for Heegaard Floer link homology

Let $T_n\subset S^3$ be the $(n,n)$-torus link. Fix a Heegaard diagram $\mathcal{D}_1=(\Sigma,\mathbf{\alpha},\mathbf{\beta},z_1,\dots,z_n,w_1,\dots,w_n)$ and a permutation $\sigma\in S_n$, and let $\mathcal{D}_2$ be obtained by permuting the basepoints according to $\sigma$. Let $F_\sigma:\widehat{HFL}(\mathcal{D}_1)\to\widehat{HFL}(\mathcal{D}_2)$ be the graded isomorphism induced by chosen diagram moves, and let $G_\sigma:\widehat{HFL}(\mathcal{D}_2)\to\widehat{HFL}(\mathcal{D}_1)$ be the canonical isomorphism induced by identifying intersection points. Symmetric-group action conjecture. The map

$$\sigma\longmapsto G_\sigma\circ F_\sigma\in\operatorname{Aut}\bigl(\widehat{HFL}(\mathcal{D}_1)\bigr)$$

defines a representation of $S_n$ on $\widehat{HFL}(\mathcal{D}_1)$. The conjecture seeks a coherent symmetric-group action from component-permuting isotopies; identifying the resulting representation, and controlling its dependence on the chosen diagram moves, remains open.