Conjecture on homogeneous convex divisible domains in flag manifolds

Let $G$ be a connected non-compact simple Lie group with trivial center, and let $P \leq G$ be a parabolic subgroup. A convex divisible domain in $G/P$ is a bounded open convex subset of an affine chart on which a discrete subgroup of $\operatorname{Aut}(\Omega)$ acts properly discontinuously, freely, and co-compactly. Assume that $G/P$ is not isomorphic to a real projective space or the complex projective plane. Homogeneity conjecture. Every convex divisible domain in $G/P$ is homogeneous. This conjecture extends known rigidity results for convex divisible domains, including Frankel's theorem in complex affine space and the classification of such domains in $\operatorname{Gr}_p(\operatorname{\mathbb{R}}^{2p})$; the general case remains open.