  Let $\mathcal{A}_g$ be the moduli space over $\overline{\mathbb{F}}_p$ of
$g$-dimensional principally polarised abelian varieties, where $p$ is a prime.
We show that if $g$ is even and $p\geq 5$, 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
$p\geq 5$ and even $g$.
