Geometric irreducibility of the next-to-non-ordinary hyperelliptic stratum

Let $k$ be a field of characteristic $p\geq 3$, and let $\mathcal{H}_g$ be the moduli space of smooth hyperelliptic curves of genus $g$ over $k$. Denote by $\mathcal{H}_g^{g-2}$ the locus of curves of $p$-rank $g-2$. $p$-rank $g-2$ conjecture. For $g\geq 3$ and $p\geq 3$, the moduli space $\mathcal{H}_g^{g-2}$ is geometrically irreducible. The claim is based on computational heuristics for lower $p$-rank strata, where smaller strata are rarer and the available evidence is weaker; it remains open.