Determine the general path–cycle relation for consecutive orders

Determine, for every integer $k\ge 5$, whether the path $P_k$ satisfies $P_k\le C_{k-1}$ in the preorder induced by rainbow forbidden subgraphs.

Background

The paper studies the preorder $H_1\le H_2$, where $H_1\le H_2$ means that every sufficiently many-colored rainbow $H_1$-free edge-colored complete graph is rainbow $H_2$-free. In the comparison between paths and cycles, the authors reduce attention to the case $k-2\le l\le k-1$ for determining whether $P_k\le C_l$. They prove that $P_k\not\le C_{k-2}$ for every $k\ge 5$.

The remaining consecutive-order case is $P_k\le C_{k-1}$. The authors establish this relation for $k=5$ and $k=6$, but explicitly state that its general validity is unknown.