Symmetric generation conjecture for the Norton algebra

Let $\omega^\perp$ be the orthogonal complement of $\omega$ in the Norton algebra $EV$, let $S$ be the subspace associated with vertices $x,y$, and let ${\rm Sym}(S)$ be its symmetric part. The Norton product is denoted by $\star$.

Symmetric generation conjecture. The subspace $\omega^\perp\cap {\rm Sym}(S)$ is the subalgebra of the Norton algebra $EV$ generated by $E\hat x+E\hat y$.

Equivalently, it should be spanned by the iterated products $B$, $B\star B$, $B\star(B\star B),\ldots$, where $B=E\hat x+E\hat y$. The supplied text gives no resolution status.