Rainbow induced-path conjecture for $K_r$-free graphs

Let $r\geq 3$, and let $G$ be a properly coloured $K_r$-free graph with chromatic number $\chi$, where $K_r$ denotes the complete graph on $r$ vertices. A rainbow induced path is an induced path whose vertices all receive distinct colours.

Rainbow induced-path conjecture. For each $r\geq 3$, every properly coloured $K_r$-free graph of chromatic number $\chi$ contains a rainbow induced path of length $\chi^{1/(r-2)}$.

This generalises the triangle-free case and is related to quantitative bounds for independent sets in $K_r$-free graphs. The source notes partial progress and that the assertion without the rainbow condition is known; the rainbow statement remains open.