Furstenberg–Katok–Spatzier–Margulis measure homogeneity conjecture

Let $n\geq 3$, let $X_n$ be the space of lattices of rank $n$, and let $H$ be a maximal $\mathbb{R}$-split torus in $\mathrm{PGL}_n(\mathbb{R})$. Let $\mu$ be an ergodic $H$-invariant probability measure on $X_n$. Furstenberg–Katok–Spatzier–Margulis conjecture. Then $\mu$ is homogeneous. The conjecture strengthens the preceding measure-classification theorem by removing the positive-entropy hypothesis. It is presented as a substantially stronger statement than the theorem available in the paper, and its resolution status is not specified in the source.