A comparison of the regularity of certain classes of monomial ideals and their integral closures

Let $S = \mathsf{k}[x_1, \ldots, x_n]$, $I$ be an ideal of $S$, and $\bar{I}$ denote its integral closure. A conjecture of Küronya and Pintye states that for any homogeneous ideal $I$ of $S$, the inequality $\operatorname{reg}(\bar{I}) \leq \operatorname{reg}(I)$ holds, where $\operatorname{reg}(\_)$ denotes the Castelnuovo-Mumford regularity. In this article, we prove the conjecture for certain classes of monomial ideals.