The projective-cover conjecture for logarithmic modules of the W_{3,2}-algebra

Let $\mathcal{W}_{3,2}$ be the logarithmic vertex operator algebra associated with the coprime pair $(3,2)$, and let $\overline{V(3,2)}$ and $\overline{MV(3,2)}$ be the logarithmic modules constructed in the paper. Let $\mathcal{W}(1)$ and $\mathcal{W}(5)$ denote the indicated irreducible $\mathcal{W}_{3,2}$-modules. Projective-cover conjecture. The logarithmic module $\overline{V(3,2)}$ is a projective cover of $\mathcal{W}(1)$, and $\overline{MV(3,2)}$ is a projective cover of $\mathcal{W}(5)$. This conjecture is motivated by agreement with the proposed structure of the projective covers in the specialized $\mathcal{W}_{3,2}$ case; the paper does not establish the projectivity claims.