The $q\to -q/t$ symmetry conjecture for one-component links

Let $\beta$ be a braid whose closure $L(\beta)$ has one connected component, written $\pi_0(L(\beta))=1$. Let $\mathrm{HH}_{m|m}(\beta)$ denote the corresponding homology, graded by variables $\mathbf{Q}$ and $\mathbf{T}$. The $q\to -q/t$ symmetry conjecture. One has

$$\mathrm{HH}_{m|m}(\beta)=\left.\mathrm{HH}_{m|m}(\beta)\right|_{\mathbf{Q}\to \mathbf{T}/\mathbf{Q}}.$$

This is presented as the analogue of the symmetry proved for HOMFLYPT homology in earlier work. The authors expect that the same method will prove it, but the conjecture remains unresolved in the source.