Long-screening kernel conjecture for principal W-algebras

Let $L=\sqrt{p}Q$ with $p\geq 2$, let $\mathfrak{g}$ be simply laced of rank $l$, and let $\mathcal{W}_k(\mathfrak{g})$ be the associated principal W-algebra with $p=k+h^\vee$. For the long screening operators $e^{\sqrt{p}\alpha_i}_0$ on the Heisenberg vertex algebra $M(1)$, the long-screening kernel conjecture.

For $p\geq 2$,

$$\mathcal{W}_k(\mathfrak{g})=\bigcap_{i=1}^l \operatorname{Ker}_{M(1)} e^{\sqrt{p}\alpha_i}_0.$$

This asserts that the long-screening realization remains valid at the positive integral, non-generic values of $p$.