Polynomial-time 3-colorability of $P_t$-free graphs with one induced odd cycle length

For any integer $t$ and any odd integer $k$, let $\mathcal{G}_{t,k}$ be the class of graphs that are $P_t$-free and whose induced odd cycles all have length $k$. Polynomial-time 3-coloring conjecture. For any fixed $t$ and $k$, the 3-coloring problem for $\mathcal{G}_{t,k}$ can be solved in polynomial time.

This claim concerns the complexity of 3-coloring a restricted graph class, motivated by results on graphs with prescribed induced cycle lengths. The source gives no resolution status for this statement.