The Hecke orbit conjecture for PEL type Shimura varieties

Let $k$ be an algebraically closed field of characteristic $p$, and let $S$ be a PEL type Shimura variety over $k$. For a point $x\in S$, let its prime-to-$p$ Hecke orbit be the orbit under prime-to-$p$ Hecke correspondences, and let the central leaf containing $x$ be the locus of points whose associated Barsotti–Tate groups have the same geometric isomorphism type as that of $x$. Hecke orbit conjecture. Every prime-to-$p$ Hecke orbit on $S$ is dense in the central leaf containing it. The conjecture describes central leaves as the minimal subvarieties stable under all prime-to-$p$ Hecke correspondences. Chai and Oort proved it for Siegel modular varieties; the stated general version for PEL type Shimura varieties remains open in general, although the paper proves it for certain irreducible components of Newton strata of PEL types A and C at unramified primes of good reduction.