The projective-cover conjecture for the constructed W_{p,p'}-modules

Let $p,p'\in\mathbb{Z}_{>0}$ with $p,p'\geq 2$ and $p$ and $p'$ relatively prime. Let $\mathcal{W}_{p,p'}$ be the logarithmic vertex operator algebra, and let $\overline{V(p,p')}$ and $\overline{MV(p,p')}$ be the logarithmic modules constructed in the paper. Consider particular irreducible $\mathcal{W}_{p,p'}$-modules whose lowest conformal weights are $1$ and $\frac{(p+2)(p'+2)}{4}$, respectively. Projective-cover conjecture. The modules $\overline{V(p,p')}$ and $\overline{MV(p,p')}$ are projective covers of those particular irreducible modules, respectively. The conjecture generalizes the proposed projective-cover interpretation in the $\mathcal{W}_{3,2}$ case. The paper constructs the modules and analyzes examples, but leaves their projectivity in general open.