Dvořák's conjecture on coloring squares of girth-five planar graphs

Let $G$ be a planar graph with girth $5$, let $\Delta$ denote its maximum degree, and let $M$ be a constant. The Dvořák conjecture. There exists some constant $M$ such that every planar graph $G$ with girth $5$ and maximum degree at least $M$ satisfies

$$\Delta(G^2)\le\Delta+2.$$

Here $G^2$ is the square of $G$ and $\Delta(G^2)$ denotes its maximum degree. This conjecture concerns the remaining high-degree, girth-five case after known results for larger girth; the supplied text gives no resolution, so it is open.