Supersingular Ekedahl-Oort strata and Oort's conjecture

Let ${\mathcal{A}}_g$ be the moduli space over ${\bar{\mathbb{F}}}_p$ of $g$-dimensional principally polarised abelian varieties, where $p$ is a prime. We show that if $g$ is even and $p\ge 5$, then every geometric generic member in the maximal supersingular Ekedahl-Oort stratum in ${\mathcal{A}}_g$ has automorphism group $\{\pm 1\}$. This confirms Oort’s conjecture in the case of $p\ge 5$ and even $g$. We also separately prove Oort’s conjecture for $g=4$ and any prime $p$.