Forest Ramsey-graph conjecture for finite ordered forest pairs

Let $H$ and $H'$ be ordered forests, and let $R_<(H,H')$ denote the family of Ramsey graphs of $(H,H')$. Forest Ramsey-graph conjecture. If $(H,H')$ is Ramsey finite, then $R_<(H,H')$ contains a forest. The claim removes a previously used $\chi$-unavoidability assumption; the source cites supporting evidence, but also states that its reverse is false, and does not resolve this direction.