Quadrics base locus conjecture for exceptional dimensions

Let $n\in\{8,16\}$, and let $L\subset\mathbb{R}_{sym}^{n\times n}$ be a subspace satisfying

$$\boldsymbol{\mathrm{P}}(L)\cap\boldsymbol{\mathrm{Y}}=\varnothing$$

and such that $\boldsymbol{\mathrm{P}}(L^\vee)\cap\boldsymbol{\mathrm{Z}}$ exists and can span $\boldsymbol{\mathrm{P}}(L^\vee)$. Quadrics base locus conjecture. The maximum dimension of $L$ in Theorem is $\frac{1}{2}n$.

The conjecture concerns the unsettled exceptional dimensions $n=8,16$ in the paper's algebraic-geometric decomposition problem. The cases $n=2,4$ are stated to be known, while the optimal dimension remains undetermined for $n=8,16$.