Projective-cover conjecture for typical and atypical ghost modules

Let $\mathscr{C}$ be the abelian category of ghost vertex operator algebra modules generated by imposing closure under extensions from the typical modules $\sigma^{\ell} \bigl( \mathcal{W}_{\lambda} \bigr)$ and the simple atypical modules $\sigma^{\ell} \bigl( \mathcal{V} \bigr)$, with the vacuum acting as the identity on these extensions. Projective-cover conjecture. In $\mathscr{C}$, the typical module $\sigma^{\ell} \bigl( \mathcal{W}_{\lambda} \bigr)$ is simple and projective, whereas the staggered module $\sigma^{\ell} \bigl( \mathcal{P} \bigr)$ is the projective cover of the simple atypical module $\sigma^{\ell} \bigl( \mathcal{V} \bigr)$.

This conjecture proposes the projective structure of the category generated by the typical and simple atypical modules. The source presents it as a conjectural description, and the projective-cover assertions remain open there.