  This paper concerns the characteristic-$p$ fibers of $\mathsf{GU}(q-2,2)$
Shimura varieties, which classify abelian varieties with additional structure.
These Shimura varieties admit two stratifications of interest: the Ekedahl-Oort
stratification, based on the isomorphism class of the $p$-torsion subgroup
scheme, and the Newton stratification, based on the isogeny class of the
$p$-divisible group. It is natural to ask which Ekedahl-Oort strata intersect
the unique closed Newton stratum, called the supersingular locus. In this
paper, we present several novel techniques that give information about the
interaction between the two stratifications for general signature $(q-2,2)$,
and as an application, we completely answer this question for the signature
$(3,2)$.
