Kobayashi's classification conjecture for irreducible complex symmetric spaces

Let $G_\mathbb{C}/K_\mathbb{C}$ be an irreducible complex symmetric space, where $G_\mathbb{C}$ is a complex semisimple Lie group and $K_\mathbb{C}$ is the fixed-point subgroup of an involutive holomorphic automorphism, up to the usual finite-component qualification. Kobayashi's classification conjecture. The space $G_\mathbb{C}/K_\mathbb{C}$ admits a compact Clifford–Klein form if and only if it is locally isomorphic to either $SO(8,\mathbb{C})/SO(7,\mathbb{C})$ or a group manifold. The conjecture extends the known examples: Riemannian symmetric spaces and reductive group manifolds admit compact Clifford–Klein forms, while the complex-sphere case predicts the exceptional dimension $7$.