  Given a family of abelian covers of $\mathbb{P}^1$ and a prime $p$ of good
reduction, by considering the associated Deligne--Mostow Shimura variety, we
obtain lower bounds for the Ekedahl-Oort type, and the Newton polygon, at $p$
of the curves in the family. In this paper, we investigate whether such lower
bounds are sharp. In particular, we prove sharpness when the number of
branching points is at most five and $p$ sufficiently large. Our result is a
generalization under stricter assumptions of [2, Theorem 6.1] by Bouw, which
proves the analogous statement for the $p$-rank, and it relies on the notion of
Hasse-Witt triple introduced by Moonen in [9].
