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

Let $S=\mathsf{k}[x_1,\dots ,x_n]$, $I$ 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 $\mathrm{reg}(\bar{I})\le \mathrm{reg}(I)$ holds, where $\mathrm{reg}(\mathrm{\_})$ denotes the Castelnuovo-Mumford regularity. In this article, we prove the conjecture for certain classes of monomial ideals.