Chai–Oort Hecke orbit conjecture for Hodge type Shimura varieties

Let $\mathcal{S}$ be the canonical integral model of a Shimura variety of Hodge type with hyperspecial level, and let $\mathcal{S}_{\overline{\mathbb{F}}_p}$ be its special fiber. A point is $\mu$-ordinary when it lies in the open Newton stratum of $\mathcal{S}_{\overline{\mathbb{F}}_p}$; this agrees with being ordinary when the ordinary locus is nonempty. Chai–Oort conjecture. The prime-to-$p$ Hecke orbit of a $\mu$-ordinary point is Zariski dense in $\mathcal{S}_{\overline{\mathbb{F}}_p}$. The conjecture was made for PEL Shimura varieties and is expected to extend to Hodge type Shimura varieties, including the PEL case; its status in the stated generality is not resolved here.