Finiteness conjecture for periodic diagonal orbits in compact sets

Let $n \geq 3$ and let $H$ be the diagonal subgroup acting on $\operatorname{PGL}_n(\mathbb{Z}) \backslash \operatorname{PGL}_n(\mathbb{R})$. Finiteness conjecture for periodic orbits. For every compact subset $\Omega$ of

$$\operatorname{PGL}_n(\mathbb{Z}) \backslash \operatorname{PGL}_n(\mathbb{R}),$$

there are only finitely many periodic $H$-orbits contained in $\Omega$. The source says this follows from the bounded-orbit conjecture together with isolation results, so its conjectural status is tied to that higher-rank rigidity framework.