Luna's conjecture for spherical subgroups of Kac–Moody groups

Let ${\mathcal H}\subseteq{\mathcal G}$ be a spherical subgroup of finite type, and associate to it the homogeneous spherical datum of ${\mathcal G}/{\mathcal H}$. Homogeneous spherical data of finite type of ${\mathcal G}$ are the data arising from such subgroups.

Luna's conjecture. Mapping ${\mathcal H}\subseteq{\mathcal G}$ to the homogeneous spherical datum of ${\mathcal G}/{\mathcal H}$ induces a bijection between the set of conjugacy classes of spherical subgroups of finite type of ${\mathcal G}$ and homogeneous spherical data of finite type of ${\mathcal G}$.

This asks whether the classification of spherical subgroups by homogeneous spherical data, known in the classical finite-dimensional theory as the Luna conjecture, extends to the Kac–Moody setting. The source presents this as an open question for groups that may be infinite-dimensional.