Rotator–Connes differential conjecture for derived annular Khovanov–Rozansky invariants

Let $\operatorname{Tr}$ be the universal dg monoidal trace, let $\operatorname{Tr}(\mathbb{1})$ be the trace of the monoidal unit, and let $X$ be a twisted complex built out of summands of $\operatorname{Tr}(\mathbb{1})$. Let $w_X$ be the rotator action on $X$, and let $\mathbb{B}$ be the Connes differential acting on the twisted complex. Rotator–Connes conjecture. The action of $w_X$ on $X$ is homotopic to

$$\operatorname{Id}+\mathbb{B}+\text{higher order terms}.$$

This predicts that annular rotation is reflected by the Connes differential in the derived annular Khovanov–Rozansky framework; the excerpt gives no resolution.