Grytczuk's bounded Thue number conjecture for planar graphs

For a graph $G$, a nonrepetitive coloring is a coloring $f:V(G)\to\mathbb N$ with no repetition in the color sequence of any simple path in $G$. The minimum number of colors in such a coloring is the Thue number $pi(G)$.

Grytczuk's conjecture. There is a constant $c$ such that

$$\pi(G)\leqslant c$$

for all planar graphs $G$. This conjecture asks whether planar graphs have uniformly bounded Thue number, extending the known boundedness results for trees, outerplanar graphs, and graphs of bounded tree-width. Its resolution is not indicated in the source.