The linear-logarithmic induced-path conjecture for triangle-free graphs

Let $G$ be a triangle-free graph with chromatic number $\chi$. An induced path is a path whose vertices induce exactly the edges of the path.

Linear-logarithmic induced-path conjecture. There is some constant $c>0$ such that every triangle-free graph of chromatic number $\chi$ contains an induced path of length at least $c\chi\log\chi$.

This is a quantitative strengthening of the known fact that triangle-free graphs of chromatic number $\chi$ contain induced paths of length at least $\chi$. The supplied status evidence concerns a different Gyárfás conjecture, so this conjecture is retained as open.