Pivotal symmetric ideal generated by the recursive idempotent

Let $\mathsf{D}_{{\mathfrak S}(n)}$ be the diagrammatic representation-theoretic object associated with the symmetric group ${\mathfrak S}(n)$, and let $\mathsf{P}^{(n)}$ be its ideal. Let $E(n+1)$ be the recursively defined idempotent from the preceding construction. Pivotal ideal conjecture. The ideal $\mathsf{P}^{(n)}$ of $\mathsf{D}_{{\mathfrak S}(n)}$ is the pivotal symmetric ideal generated by $E(n+1)$. This identifies the relevant ideal through a single recursively defined idempotent and connects the diagrammatic ideal structure with the representation theory of the symmetric group. The source gives no evidence of a resolution.