NP-completeness of deciding path-chromatic number at most two

For a graph $G$, let $\mathsf{path}\textnormal{-}\chi(G)$ denote its path-chromatic number. Path-chromatic two-colourability conjecture. It is NP-complete to decide if $\mathsf{path}\textnormal{-}\chi(G) \leqslant 2$. The source states this alongside the tree-chromatic claim, but does not explicitly introduce it with the phrase “we conjecture”; the surrounding context presents it as part of the same unresolved complexity question.