Finiteness and non-semisimplicity conjecture for modules of ${\mathcal V}^{(p)}$

For $p\geq 3$, let ${\mathcal V}^{(p)}$ be the generalized vertex algebra introduced by

$${\mathcal V}^{(p)}=\operatorname{Ker}_{M\otimes F_{p/2}}\widetilde{Q}.$$

Consider its modules in the category $\mathcal{O}$. Module-category conjecture. For every $p\geq 3$, ${\mathcal V}^{(p)}$ has finitely many irreducible modules in the category $\mathcal{O}$, and there exist non-semisimple ${\mathcal V}^{(p)}$-modules in that category. This is motivated by the case $p=2$ and predicts both finite irreducible-module type and non-semisimple representation theory for the higher members of the family.