Irreducibility conjecture for associated varieties with finitely many symplectic leaves

Let $V$ be a simple, finitely strongly generated, positively graded conformal vertex operator algebra with $V_0=\mathbb{C}$, and let $X_V=\operatorname{Specm}(R_V)$ be its associated variety. Assume that $X_V$ has finitely many symplectic leaves. Irreducibility conjecture. The associated variety $X_V$ is irreducible. This conjecture concerns the geometry of associated varieties of vertex operator algebras; the paper proves it in new cases, but the general assertion under these hypotheses remains open.