Coset rationality and lisse-ness conjecture

Let $V$ be a vertex algebra and let $W$ be a vertex subalgebra of $V$. Suppose that $V$ and $W$ are finitely strongly generated, conformal, rational, and lisse. Write

$$\operatorname{Com}(W,V)=\{w\in V\mid [v_{(m)},w_{(n)}]=0\text{ for all }m,n\in\mathbb{Z},\ v\in W\}.$$

Coset conjecture. The commutant $\operatorname{Com}(W,V)$ is also rational and lisse. This is presented as a widely believed assertion and is used in the paper to deduce rationality and lisse-ness of certain principal W-algebras; its general validity remains open.