The nilpotent-rank conjecture for finitely generated 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 vertex operator algebra defined as the intersection of the kernels of the screening operators $Q$ and $\widetilde{Q}$, and let $M$ be a finitely generated $\mathcal{W}_{p,p'}$-module. The $L(0)$-nilpotent rank of $M$ is the largest Jordan-block size of $L(0)$, equivalently the least $k$ such that $(L(0)-L_{ss}(0))^k=0$ on $M$. Nilpotent-rank conjecture. Every finitely generated $\mathcal{W}_{p,p'}$-module has $L(0)$-nilpotent rank at most $3$. The claim is motivated by conjectural character formulas for irreducible $\mathcal{W}_{p,p'}$-modules. The paper gives constructions with rank three, but does not prove the asserted bound for all finitely generated modules.