The vanishing-cycle W-algebra action conjecture for framed quiver moduli spaces

Let $\mathcal Q(r,n)$ be the framed quiver moduli space, and let $H_{{\bf T}_0}^{\sf van}(\mathcal Q(r,n))$ denote the dual of its compactly supported equivariant vanishing-cycle cohomology. Consider the direct sum

$$\bigoplus_{n\geq 0} H_{{\bf T}_0}^{\sf van}(\mathcal Q(r,n)).$$

Vanishing-cycle W-algebra action conjecture. There is a $W_\kappa({\mathfrak gl}_r)$-action on this direct sum identifying it with the $W$-vacuum module.

This conjecture extends the established $W$-algebra action on the Hilbert-scheme-related moduli spaces to the more natural Calabi–Yau framed quiver moduli spaces, which admit a global critical-locus presentation and an equivariant sheaf of vanishing cycles. The source presents this as an open direction, and no resolution is supplied here.