The induced rainbow path conjecture for triangle-free graphs

Let $G$ be a finite simple triangle-free graph, let $\chi(G)$ be its chromatic number, and let $\phi:V(G)\to[k]$ be a proper vertex coloring, where $k=\chi(G)$. A path in $G$ is rainbow if all its vertices have distinct colors under $\phi$, and induced if no two nonconsecutive vertices of the path are adjacent in $G$. Induced rainbow path conjecture. Every colored triangle-free graph $(G,\phi)$ contains an induced rainbow path on $\chi(G)$ vertices. This would simultaneously strengthen the known existence of induced paths and rainbow paths of order equal to the chromatic number in colored triangle-free graphs. The conjecture remains open in general, although several partial results are known.