A counterexample to a conjecture of Küronya and Pintye on regularity and integral closure

We exhibit an equigenerated monomial ideal $I\subseteq K[x,y,z,w]$ with $\mathrm{reg}(\bar{I})>\mathrm{reg}(I)$. The ideal $I$ is generated in degree 4 and satisfies $\mathrm{reg}(I)=4$, while its integral closure $\bar{I}$ has a minimal generator of degree 5 and satisfies $\mathrm{reg}(\bar{I})=5$. This gives a counterexample to the polynomial-ring formulation of the Küronya–Pintye conjecture.