  We exhibit a class of Hibi rings which are diagonally F-regular over fields
of positive characteristic, and diagonally $F$-regular type over fields of
characteristic zero, in the sense of Carvajal-Rojas and Smolkin. It follows
that such Hibi rings satisfy the uniform symbolic topology property effectively
in all characteristics. Namely, for rings $R$ in this class of Hibi rings, we
have $P^{(dn)} \subseteq P^n$ for all $P \in \operatorname{Spec} R$, where $d =
\dim(R)$. Further, we demonstrate that all Hibi rings over fields of positive
characteristic are 2-diagonally $F$-regular, and that the simplest Hibi ring
not contained in the above class is not 3-diagonally $F$-regular in any
characteristic. The former implies that $P^{(2d)} \subseteq P^2$ for all $P \in
\operatorname{Spec} R$.
