Non-generation conjecture for higher-dimensional polar Grassmannians

Let $\mathbb{F}_0$ be a proper subfield of a field $\mathbb{F}$, and let $Q^+_{n-1}(2n-1,\mathbb{F})$ denote the polar Grassmannian of maximal singular subspaces of the hyperbolic quadric $Q^+(2n-1,\mathbb{F})$. Say that this polar Grassmannian is $\mathbb{F}_0$-generated when it is generated by its $\mathbb{F}_0$-rational elements. Non-generation conjecture. For every $n>2$ and every proper subfield $\mathbb{F}_0$ of $\mathbb{F}$, $Q^+_{n-1}(2n-1,\mathbb{F})$ is never $\mathbb{F}_0$-generated. The conjecture is motivated by the proof of the corresponding result in the case of lines on $Q^+(5,\mathbb{F})$, which uses that each such line lies in precisely two singular planes; its validity in all higher dimensions remains open.