Symmetric-subspace closure conjecture for the Norton algebra

Let $S$ be the subspace associated with vertices $x,y$ in the Norton algebra $EV$, and let ${\rm Sym}(S)$ denote its symmetric part under interchange of $x$ and $y$. The product is the Norton product $\star$.

Symmetric-subspace closure conjecture. We have

$${\rm Sym}(S)\star {\rm Sym}(S)\subseteq {\rm Sym}(S).$$

This asserts that the symmetric part of $S$ is a subalgebra under the Norton product. The supplied text gives no resolution status.