Quadrics base locus conjecture for normed-division-algebra dimensions

For $n\in\{2,4,8,16\}$, let $W_{n,\operatorname{sym}}$ be a linear subspace of $\mathbb{R}_{\operatorname{sym}}^{n\times n}$ with

$$\dim(W_{n,\operatorname{sym}})=\frac{1}{2}n+1,$$

and suppose every nonzero element of $W_{n,\operatorname{sym}}$ is invertible. Let $\{A_1,\ldots,A_{\frac{1}{2}n+1}\}$ be any basis of $W_{n,\operatorname{sym}}$. Quadrics base locus conjecture. The system

$$\begin{cases} \mathbf{x}^T A_1\mathbf{x}=0\\ \cdots\\ \mathbf{x}^T A_{\frac{1}{2}n+1}\mathbf{x}=0 \end{cases}$$

has no solution for $\mathbf{x}\in\mathbb{R}^n\setminus\{0\}$.

This conjecture extends the limitation phenomenon observed computationally for $n=2$ to the exceptional dimensions $n=4,8,16$, which are associated with the real normed division algebras. The supplied passage presents it as motivated by computational experiments and historical connections; no resolution is given.