High-girth Ramsey graph conjecture for ordered forests

Let $H$ and $H'$ be ordered forests, and let $t$ be a positive integer. Write $R_<(H,H')$ for the family of Ramsey graphs of $(H,H')$, and let $\operatorname{girth}(F)$ denote the girth of a graph $F$. High-girth Ramsey graph conjecture. There is an $F\in R_<(H,H')$ such that

$$\operatorname{girth}(F)\geq t.$$

This would extend the corresponding high-girth Ramsey-graph property from unordered forests to ordered forests; the paper presents it as an expected fact, and no resolution is given.