The Thue list-number conjecture for the infinite path

A graph coloring is nonrepetitive if the color sequence along every path contains no square. The Thue-list number of a graph is the least integer $k$ such that, whenever every vertex has a list of at least $k$ available colors, the graph has a nonrepetitive coloring choosing each vertex's color from its list.

Thue list-number conjecture. The Thue list number of the infinite path is $3$.

This is a list-coloring analogue of Thue's results on avoiding repetitions in words. The source presents it as a challenging conjecture and considers a word-theoretic reformulation; no resolution is given here.