The CDSW conjecture for simple Lie algebras

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$, let $g$ be its dual Coxeter number, and let $A$ be the quotient of $R=\wedge(\mathfrak{g}\oplus\mathfrak{g})$ by the ideal generated by the three canonical copies of $\mathfrak{g}$ in bidegrees $(2,0)$, $(1,1)$, and $(0,2)$. Let $S$ be the unique invariant element of $A$ of degree $(1,1)$, namely $S=\operatorname{Tr}|_V(XY)$ for a non-trivial irreducible finite-dimensional representation $V$ of $\mathfrak{g}$. The CDSW conjecture. The invariant algebra $A^{\mathfrak{g}}$ is generated by $S$, and

$$S^g=0,\qquad S^{g-1}\ne 0.$$

The conjecture was known for classical Lie algebras and was established in the paper's stated cases, including $G_2$; the source presents the three assertions as the conjecture motivating the paper's further conjectures.