Küronya–Pintye conjecture on regularity and integral closure

Let $R=k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$, and let $I$ be a homogeneous ideal of $R$. Write $\overline{I}$ for the integral closure of $I$ and $\operatorname{reg}(I)$ for its Castelnuovo–Mumford regularity.

Küronya–Pintye conjecture.

$$\operatorname{reg}(\overline{I})\leq \operatorname{reg}(I).$$

The conjecture concerns whether taking integral closure can increase Castelnuovo–Mumford regularity. The paper gives a positive answer for dominant monomial ideals and almost complete intersection monomial ideals, while the general homogeneous case remains open in the supplied context.