The basis and generation conjecture for the Norton subalgebra of the bilinear forms graph

Let $\Gamma=(X,\mathcal R)$ be the bilinear forms graph, let $x,y\in X$ be adjacent vertices, and let $U=U(x,y)$ be the subspace defined earlier. Let $E=E_1$ be the primitive idempotent, and let $\widehat O^{(B)}_{1,1}$, $\widehat O^{(C)}_{1,1}$, and $\widehat O^{(D)}_{1,1}$ denote the indicated vectors associated with the corresponding cells of the equitable partition. The Norton product on $EV$ is $u\star v=E(u\circ v)$. Basis and generation conjecture. The following hold:

(i) The subspace $EU$ has basis

$$E\hat x,\qquad E\hat y,\qquad E\widehat O^{(B)}_{1,1},\qquad E\widehat O^{(C)}_{1,1},\qquad E\widehat O^{(D)}_{1,1}.$$

(ii) The Norton subalgebra $EU$ is generated by $E\hat x$ and $E\hat y$.

This conjecture concerns the structure of the Norton algebra associated with the $Q$-polynomial bilinear forms graph. Establishing the proposed basis and generation statement would give an explicit description of the Norton subalgebra determined by two adjacent vertices.