The exceptional Lie algebra trace-generation conjecture

Let $G$ be one of the exceptional Lie groups, let $(V,\pi)$ be respectively the $7$-, $26$-, $27$-, $56$-, or $248$-dimensional representation for $G_2$, $F_4$, $E_6$, $E_7$, or $E_8$, and let $T(\mathfrak{g})^G$ denote the algebra of $G$-invariants in the tensor algebra of the Lie algebra $\mathfrak{g}$. Traces are defined using the representation $(V,\pi)$ as in the preceding discussion. Exceptional trace-generation conjecture. For these choices of $G$ and $(V,\pi)$, $T(\mathfrak{g})^G$ is generated by traces, allowing for permutation of the indices. The conjecture proposes that the trace description established for the classical groups extends to the exceptional Lie algebras; its status is not determined in the supplied source context.