Premet's one-dimensional representation conjecture for W-algebras

Let $\mathcal W$ be a W-algebra, and call a one-dimensional representation a one-dimensional $\mathcal W$-module; equivalently, such a representation corresponds to a two-sided ideal of codimension $1$.

Premet's conjecture. Every W-algebra has a one-dimensional representation, equivalently a two-sided ideal of codimension $1$.

The conjecture concerns the existence of one-dimensional representations for arbitrary W-algebras. The supplied text states that it was made by Premet and records that it was known to be true except for several cases in type $E_8$, where it remained open at the time of writing.