Generic p-torsion conjecture for smooth and hyperelliptic curves

Let $0\leq f<g$, let ${\cal M}_g$ be the moduli space of smooth curves of genus $g$, and let ${\cal H}_g$ be the hyperelliptic locus when $p\geq3$. Write $V_{g,f}$ for the locus of principally polarized abelian varieties of dimension $g$ and $p$-rank at most or equal to $f$, and let $I_{g-f}$ be the unique symmetric ${\rm BT}_1$ group scheme of rank $p^{2(g-f)}$, $p$-rank $0$, and $a$-number $1$. Generic p-torsion conjecture. If $X$ is the generic point of any component of $V_{g,f}\cap{\cal M}_g$ (respectively, $V_{g,f}\cap{\cal H}_g$ for $p\geq3$), then

$$J_X[p]\cong({\mathbb Z}/p\oplus\mu_p)^f\oplus I_{g-f}.$$

Equivalently, a generic smooth curve (and, for $p\geq3$, a generic hyperelliptic curve) of genus $g$ and $p$-rank $f$ has $a$-number $1$. The conjecture refines the known stratification by $p$-rank by predicting the generic isomorphism class of the Jacobian's $p$-torsion, including on the hyperelliptic locus.