Supersingular curves of genus four in characteristic two

We describe the intersection of the Torelli locus $j(\mathcal{M}_4^{ct}) = \mathcal{J}_4$ with Newton and Ekedahl-Oort strata related to the supersingular locus in characteristic two. We show that the locus of supersingular Jacobians $\mathcal{S}_4\cap\mathcal{J}_4$ in characteristic two is pure of dimension three. One way to obtain that result uses an analysis of the data of smooth genus four curves and principally polarized abelian fourfolds defined over $\mathbb{F}_2$, and another involves studying the Ekedahl-Oort types and using the indecomposability of some $\mathrm{BT}_1$ group schemes.