The MLDE conjecture for characters of the vertex algebras $\mathcal{C}_n$

Let $\mathcal{C}_n$ be the vertex algebra under consideration, and let

$$\operatorname{ch}[\mathcal{C}_n]$$

be its character. An MLDE is a modular linear differential equation of the form described using Serre derivatives, with coefficients in the graded ring of modular forms. Character MLDE conjecture. For $n\geq 2$, the character $\operatorname{ch}[\mathcal{C}_n]$ satisfies an MLDE with holomorphic coefficients in $M_*(\Gamma(1))$ when $n$ is odd and in $M_*(\Gamma(2))$ when $n$ is even. This is presented as a consequence that would follow if $\mathcal{C}_n$ were quasi-lisse; the supplied text does not establish quasi-lisse-ness or otherwise resolve the claim.