Grytczuk's conjecture on bounded nonrepetitive chromatic number of planar graphs

Let $G$ be a graph. A vertex coloring is nonrepetitive if no path has a color sequence of the form $s_1,s_2,\dots,s_t,s_1,s_2,\dots,s_t$. Let $\pi(G)$ denote the minimum number of colors in a nonrepetitive vertex coloring of $G$. Grytczuk's conjecture. There is an absolute constant $K$ such that every planar graph $G$ satisfies

$$\pi(G)\le K.$$

This is identified in the source as a major open problem; the paper resolves the analogous facial-coloring conjecture but does not state a resolution of Grytczuk's planar-graph conjecture.