Limbeek–Zimmer's reformulated conjecture for almost-homogeneous domains

Let $G$ be a simple Lie group and let $P$ be a parabolic subgroup of $G$. A domain in $G/P$ is proper almost-homogeneous if it is proper and admits an automorphism group with a compact quotient. Limbeek–Zimmer's reformulated conjecture.

1. If $G/P$ admits proper almost-homogeneous domains, then it is an irreducible Nagano space.
2. If $G/P$ is an irreducible Nagano space $N$, and if it is neither Example (ix) nor Example (iv) with $\min(p,q)=1$ in Table~, then any proper almost-homogeneous domain of $N$ is a realization of $\mathbb{X}(N)$, and is in particular symmetric.

The conjecture refines the preceding Limbeek–Zimmer conjecture. The paper proves some cases and reports that the analogous assertion for complex Nagano spaces is proved in forthcoming work, but the full statement remains open.