Matsuki–Gindikin conjecture on domains associated with flag-manifold orbits

Let $G_\mathbb C$ be a connected complex semisimple Lie group, let $G_\mathbb R$ be a connected real form, and let $K_\mathbb C$ be the complexification of a maximal compact subgroup of $G_\mathbb R$. Let $X=G_\mathbb C/P$ be a flag manifold that is not $K_\mathbb C$-homogeneous. For each $K_\mathbb C$-orbit $S$ on $X$, let $C(S)$ be the associated $G_\mathbb R$-$K_\mathbb C$-invariant subset of $G_\mathbb C$, and let $C(S)_0$ denote its connected component containing the identity. Let $D$ be the Akhiezer–Gindikin domain.

Matsuki–Gindikin conjecture. For every $K_\mathbb C$-orbit $S$ of nonholomorphic type on $X$, one has

$$C(S)_0=D.$$

The conjecture concerns the equivalence of domains arising from duality between $K_\mathbb C$- and $G_\mathbb R$-orbits on flag manifolds. The source states that it was already proved for closed and open orbits, for the remaining orbits when $G_\mathbb R$ is of non-Hermitian type, and, in this paper, for arbitrary non-closed orbits when $G_\mathbb R$ is of Hermitian type; thus it is completely solved affirmatively.