The idempotent criterion conjecture for Mathieu-Zhao subspaces of vertex operator algebras

Let $V$ be a $\mathbb Z$-graded vertex operator algebra with unique primitive idempotents $\{e_1,\dots,e_m\}$, and write

$$V=\bigoplus_{n\in\mathbb Z}V_n.$$

Suppose that $V$ decomposes into ideals

$$V=V^1\oplus\cdots\oplus V^m,$$

where

$$V^i=e_i(-1)V.$$

The idempotent criterion conjecture. A subspace $M\subseteq V$ is a Mathieu–Zhao subspace of $V$ if and only if, whenever $e_i\in M$, one has $V^i\subseteq M$. This is proposed as the vertex-algebra analogue of the idempotent criterion for associative algebras; the source does not establish the converse.