Refinement of the supersingular Ekedahl–Oort stratification by the J-stratification

Prove that the stratification introduced by Karemaker and Yu on the union of all supersingular Ekedahl–Oort strata refines the J-stratification of the affine Deligne–Lusztig variety associated with the supersingular locus.

Background

The paper studies the supersingular locus of the moduli space of principally polarized abelian varieties through affine Deligne–Lusztig varieties and their J-stratifications. The J-stratification records relative-position data under the action of the group of self-quasi-isogenies of a superspecial principally polarized p-divisible group.

Karemaker and Yu introduced a separate stratification of the union of all supersingular Ekedahl–Oort strata, whose unique maximal stratum is open and dense in that union. The paper notes that this stratification would provide a finer geometric organization of the supersingular Ekedahl–Oort strata if it refines the J-stratification, but explicitly states that this refinement was only conjectured.