Etingof's degree-$g$ relation conjecture for the \widehat p_i

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra with dual Coxeter number $g$. Let $p_1,\ldots,p_r$ be homogeneous generators of $(S\mathfrak{g})^{\mathfrak{g}}$, and let $\widehat p_i$ be the corresponding elements of $E=(B\otimes B)^{\mathfrak{g}}$. Etingof's degree-$g$ relation conjecture. There exists a relation of degree $g$ among the $\widehat p_i$ that contains the term $\widehat p_1^g$ with a nonzero coefficient. This conjecture is proposed as an implication toward the nilpotence assertion in the CDSW conjecture; the supplied source context does not establish it in general.