MLDE conjecture for twisted characters of DC-series affine vertex algebras

Let $V$ be a simple affine vertex algebra associated with one of the Lie algebras in the $\mathrm{DC}$ series

$$A_{1}\subset A_{2} \subset G_{2} \subset D_{4} \subset F_{4}\subset E_{6} \subset E_{7} \subset E_{8}$$

at level $-h^{\vee}/6-1$. Let $\chi^{\mathrm{twi}}(q)$ denote a normalized $\mathbb{Z}_{2}$-twisted character. MLDE conjecture. The functions $\chi^{\mathrm{twi}}(q)$ are solutions of a second-order $\Gamma^{0}(2)$-MLDE with suitable coefficients $a_{1}$ and $a_{2}$, without a $D^{(1)}_{q}$ term:

$$\left(D^{(2)}_{q}+a_{1}\Theta_{0,2}+a_{2}\Theta_{1,1}\right)\chi^{\mathrm{twi}}(q)=0.$$

This conjecture predicts a uniform second-order modular linear differential equation for normalized characters of the $\mathbb{Z}_{2}$-twisted modules across the DC series; the paper establishes analogous MLDE results in several examples, while the general assertion remains open.