Dense-orbit and finite-orbit equivalence for orthogonal and symplectic Grassmannians

Let $G$ be a simple connected irreducible subgroup of either $SO(V)$ or $Sp(V)$, where $V$ is the underlying orthogonal or symplectic $G$-module, and let $d4a2_k(V)$ denote the corresponding orthogonal or symplectic Grassmannian of $k$-dimensional subspaces. Let $p$ be the characteristic parameter appearing in the exceptional case.

Dense-orbit and finite-orbit equivalence. Unless

$$(G,V,p,k)=(C_3,\lambda_2,p\neq 3,k=7),$$

the action of $G$ on

$$\mathcal{S}_k(V)$$

has a dense orbit if and only if $G$ acts on $\mathcal{S}_k(V)$ with finitely many orbits.

This conjecture extends the known equivalence for ordinary Grassmannians and its established analogues for self-dual modules in the cases $\mathcal{S}_1(V)$ and $\mathcal{S}_2(V)$. The stated $C_3$ case is the sole proposed exception; the broader equivalence for orthogonal and symplectic Grassmannians remains open.