Supersingular Ekedahl-Oort strata and Oort's conjecture

Let Ag be the moduli space over F̄ₚ of g-dimensional principally polarised abelian varieties, where p is a prime. We show that if g is even and p≥ 5, then every geometric generic member in the maximal supersingular Ekedahl-Oort stratum in Ag has automorphism group \ ± 1\. This confirms Oort's conjecture in the case of p≥ 5 and even g.