The Weyl embedding conjecture for polar Grassmannians

Let $\mathbb{F}$ be a field, let $\widehat{\mathbb{F}}$ be the chosen minimal-degree separable quadratic extension in the case $d_0=2$, and write

$$\widehat{\mathcal{Q}}_n=\mathcal{Q}_n(n+1,0,0;\widehat{\mathbb{F}}),\qquad \widetilde{\mathcal{Q}}_n=\mathcal{Q}_n(n,1,0;\widehat{\mathbb{F}}).$$

Let $\widehat{e}^W_n$ be the Weyl embedding of $\widehat{\mathcal{Q}}_n$, let $\widetilde{e}^W_n$ be the canonical veronesean embedding of $\widetilde{\mathcal{Q}}_n$, and let $e^W_n$ be the Weyl-like embedding of $\mathcal{Q}_n$ with codomain $V^W_k$. Let $e^{\mathrm{spin}}_n$ denote the spin-like embedding of $\mathcal{Q}_n$, and let $\nu$ be the canonical veronesean embedding of $\mathrm{PG}(2^n-1,\mathbb{F})$ in $\mathrm{PG}({{2^n+1}\choose 2}-1,\mathbb{F})$.

The Weyl embedding conjecture. The Weyl embedding $\widehat{e}^W_n$ of $\widehat{\mathcal{Q}}_n$ induces on $\widetilde{\mathcal{Q}}_n$ its veronesean embedding $\widetilde{e}^W_n$, the codomain $V^W_k$ of $e^W_n$ is the canonical Baer subgeometry of $\mathrm{PG}({{2^n+1}\choose 2}-1,\widehat{\mathbb{F}})$ defined over $\mathbb{F}$, and $\widetilde{e}^W_n$ induces $e^W_n$ on $\mathcal{Q}_n$. Consequently,

$$e^W_n=\nu\cdot e^{\mathrm{spin}}_n.$$

In particular, if $e^W_n=\varepsilon_n$, as when $\operatorname{char}(\mathbb{F})\neq 2$, then

$$\varepsilon_n=\nu\cdot e^{\mathrm{spin}}_n.$$

The conjecture concerns the compatibility of the Weyl, spin-like, and veronesean embeddings under the inclusions of these polar Grassmannians; the statement is presented as an unproved expectation, and no resolution is supplied in the source.