Geometric irreducibility of the non-ordinary hyperelliptic locus

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 $k$. Write $\mathcal{H}_g^{g-1}$ for its non-ordinary, or $p$-rank $g-1$, locus. Non-ordinary-locus conjecture. For $g\geq 3$ and $p\geq 3$, the locus $\mathcal{H}_g^{g-1}$ is geometrically irreducible. This extends the known fact that $\mathcal{H}_2^1$ is geometrically irreducible; the conjecture is supported by computational sampling but remains open in the stated generality.