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Given a family of abelian covers of $\mathbb{P}^1$ and a prime $p$ of goodreduction, by considering the associated Deligne--Mostow Shimura variety, weobtain 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 lowerbounds are sharp. In particular, we prove sharpness when the number ofbranching points is at most five and $p$ sufficiently large. Our result is ageneralization under stricter assumptions of [2, Theorem 6.1] by Bouw, whichproves the analogous statement for the $p$-rank, and it relies on the notion ofHasse-Witt triple introduced by Moonen in [9].

Abelian covers of $\mathbb{P}^1$ of $p$-ordinary Ekedahl-Oort type