The conjecture that the ideal $L$ is generated by the higher \widehat p_i

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra, let $E=(B\otimes B)^{\mathfrak{g}}$, and let $p_1,\ldots,p_r$ and $\widehat p_1,\ldots,\widehat p_r$ be as above. Let $L$ be the ideal in $E$ used in the paper. The ideal-generation conjecture. The generation conjecture for $P=\operatorname{span}(\widehat p_1,\ldots,\widehat p_r)$ holds, and $L$ is generated by $\widehat p_2,\ldots,\widehat p_r$. The claim is presented as a further conjecture implying the first part of the CDSW conjecture; the supplied context gives no general proof or resolution.