Dvořák–Kráľ–Nejedlý–Škrekovski girth-five planar square-coloring conjecture

Let $G$ be a planar graph of girth at least $5$, and let $\Delta$ be its maximum degree. Dvořák–Kráľ–Nejedlý–Škrekovski conjecture. There exists $\Delta_0$ such that, if $\Delta\ge\Delta_0$, then

$$\chi^2(G)=\Delta+2.$$

This conjecture concerns the sharp asymptotic bound in girth five and was confirmed by Bonamy, Cranston, and Postle.