Etingof's conjecture that the elements $\widehat p_i$ generate the invariant algebra

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$, let $B$ be the algebra used in the paper, and set $E=(B\otimes B)^{\mathfrak{g}}$. Let $p_1,\ldots,p_r$ be homogeneous generators of $(S\mathfrak{g})^{\mathfrak{g}}$, with degrees $d_1=2,\ldots,d_r$, and let $\widehat p_i$ be the associated elements of $E$; write $P$ for their span. Etingof's generation conjecture. The subspace $P$ generates $E$. The conjecture holds for classical Lie algebras, as shown by the invariant-theoretic evidence given in the source.