Action of the deformed W-algebra beyond tautological classes

Let $X$ be a two-sided Hecke pattern of rank $r$. Write $H_*(X)$ for its Borel–Moore homology, and let $W_{\star,\star}(S)$ denote the algebra with the appropriate choices of indices $\star\in\{\uparrow,\downarrow\}$. Action beyond tautological classes. For an appropriate $\star\in\{\uparrow,\downarrow\}$, there is an action of $W_{\star,\star}(S)$ on $H_*(X)$. The conjecture extends the action presently constructed on tautological classes to the whole Borel–Moore homology; the relevant algebraic relations beyond the tautological subspace remain unproved.