Oort's conjecture on automorphisms of supersingular abelian varieties

Let $k$ be a field of characteristic $p>0$, let $E$ be a supersingular elliptic curve over $k$, and for $g\geq 2$ let $\mathcal{A}_g$ denote the moduli space of principally polarized abelian varieties of dimension $g$ over $\mathbb{F}_p$. Let $\mathcal{S}_g\subseteq\mathcal{A}_g$ be the closed supersingular locus. An automorphism means a geometric automorphism. Oort's conjecture. For any $g\geq 2$ and a prime number $p>0$, every component of the supersingular locus $\mathcal{S}_g\subseteq\mathcal{A}_g$ has generic automorphism group $\{\pm1\}$. This conjecture concerns the generic automorphisms of components of supersingular loci; the paper proves it for $g=4$ and $p>2$, and gives a new proof for $g=3$ and $p>2$, while the general statement remains open.