The two-dimensionality conjecture for ideals in the triplet vertex algebra

Let $p\geq 2$, let $A(\mathcal{W}(p))$ be the Zhu algebra of the triplet vertex algebra, and let $h_{i,1}$ denote the relevant conformal weights. For each $i$, let $\mathbb{I}_{h_{i,1}}$ be the ideal in $A(\mathcal{W}(p))$ spanned by $v_i$ and $w_i$. Two-dimensionality conjecture. Each $\mathbb{I}_{h_{i,1}}$ is a two-dimensional ideal. This conjecture concerns the structure of the Zhu algebra and is used to establish the expected decomposition of the algebra; the source later proves the corresponding statement for every prime $p$, while also proving it for $i=p-1$ for all $p\geq 2$.