The page-number-2 conjecture for nonrepetitive graph coloring

A graph has page number $k$ if its vertices can be arranged on a spine and its edges partitioned into $k$ pages, with no two edges on the same page crossing in the spine order. A vertex coloring is nonrepetitive if no path has a color sequence of the form $XX$ for a nonempty word $X$. Page-number-2 conjecture. There is a finite constant $N$ such that every graph of page number $2$ has a nonrepetitive coloring using $N$ colors.

This question was propounded by Idziak and is presented as a major open problem in nonrepetitive graph coloring. The paper notes that it would imply the existence of a bounded nonrepetitive coloring for planar graphs.