Equivariance conjecture for the instanton skein exact triangle

Let $L_+$, $L_-$ and $L_0$ be oriented links, with $L_0$ the resolution of a distinguished crossing, and suppose the skein exact triangle involves the module

$$\mathit{KHI}^-(L_0)\mathrel{\widetilde{\otimes}}_{\mathbb{C}[U_1,U_2]}\mathbb{C}[U_1,U_2]/(U_1-U_2).$$

If $L_0$ has $\ell$ components, let $\mathbb{C}[U_1,\dots,U_{\ell-1}]$ denote the polynomial ring in the variables associated to all but one component. Equivariance conjecture. There is a natural $\mathbb{C}[U_1,\dots,U_{\ell-1}]$-module structure on

$$\mathit{KHI}^-(L_0)\mathrel{\widetilde{\otimes}}_{\mathbb{C}[U_1,U_2]}\mathbb{C}[U_1,U_2]/(U_1-U_2)$$

so that the maps $f$, $g$, and $h$ in the skein exact triangle are $\mathbb{C}[U_1,\dots,U_{\ell-1}]$-equivariant. The conjecture would strengthen the currently established graded skein exact triangle by making its maps compatible with the natural multivariable module structure; the supplied text gives no resolution status.