Injective $(\Delta+1)$-coloring of planar graphs with girth 6

A vertex coloring of a graph $G$ is called injective if every two vertices joined by a path of length 2 get different colors. The minimum number $\chi_i(G)$ of the colors required for an injective coloring of a graph $G$ is clearly not less than the maximum degree $\Delta(G)$ of $G$. There exist planar graphs with girth $g\ge6$ and $\chi_i=\Delta+1$ for any $\Delta\ge2$. We prove that every planar graph with $\Delta\ge18$ and $g\ge6$ has $\chi_i\le\Delta+1$.