Central leaves in the special fiber of Shimura varieties are the loci where the isomorphism class of the universal $p$-divisible group remains constant. The Hecke orbit conjecture asserts that every prime-to-$p$ Hecke orbit in a PEL type Shimura variety is dense in the central leaf containing it. This conjecture is proved for Hilbert modular varieties by C.-F. Yu, and for Siegel modular varieties by Chai and Oort. In this talk I give an overview of the conjecture and present my work that generalizes Chai and Oort's strategy to irreducible components of certain Newton strata on Shimura varieties of PEL type.