The Whittaker conditions for fundamental classes of Uhlenbeck spaces

Let $|1^d\rangle=[\mathcal U_G^d]$ be the fundamental class in equivariant intersection cohomology, and let $\widetilde W^{(\kappa)}_n$ denote the positive modes of the generators of the universal W-algebra, where $1\leq \kappa\leq \ell$. Whittaker conjecture. For $d\geq1$ and $n>0$,

$$\widetilde W^{(\kappa)}_n|1^d\rangle= \begin{cases} |1^{d-1}\rangle&\text{if $\kappa=\ell$ and $n=1$},\\ 0&\text{otherwise}. \end{cases}$$

These relations characterize the fundamental classes as Whittaker-type vectors and connect the geometry of Uhlenbeck spaces with representation theory of the W-algebra. The source presents them as conjectural and gives no resolution evidence.