Unboundedness of the geometric component count of the genus-three p-rank-zero locus

Let $\mathcal{H}_3^0$ denote the moduli space of smooth hyperelliptic curves of genus $3$ and $p$-rank $0$ in characteristic $p\geq 3$. Let $c_{3,0,p}$ be the number of geometric irreducible components of $\mathcal{H}_3^0$. Genus-three component-growth conjecture. The number $c_{3,0,p}$ of geometric components of $\mathcal{H}_3^0$ tends to infinity as $p$ grows. Computational data suggests this growth, although the sampling statistics exhibit non-integral behavior for some methods and field sizes; the conjecture remains open.