López-Bracho et al.'s robust coloring conjecture for paths

Let $G$ and $H$ be two paths with $n$ edges on the same vertex set.

López-Bracho et al.'s robust coloring conjecture. There exists a $3$-coloring of $G$ such that the number of monochromatic edges of $H$ is at most

$$\left\lfloor\frac{n+1}{4}\right\rfloor.$$

This conjecture concerns robust colorings of one path against the edges of another path on the same vertex set. The supplied source does not indicate whether the conjecture has been resolved.