Higher-rank homogeneous measure classification conjecture

Let $n \geq 3$, let $G \cong \operatorname{PGL}_n(\mathbb{R})$, and let $\Gamma < G$ be a lattice as in examples (L-1) or (L-2) of the source. Let $H$ be the relevant diagonal subgroup and let $\mu$ be an $H$-invariant ergodic measure. Homogeneous measure classification conjecture. The measure $\mu$ is homogeneous: there is a closed group $L \leq G$ such that $\mu$ is an $L$-invariant measure on a single $L$-orbit. This is part of the higher-rank measure-rigidity picture, with related conjectures attributed to Furstenberg, Katok–Spatzier, and Margulis; the source presents it as expected and does not indicate a resolution.