Super-polynomial directed Ramsey numbers with three colors

Let $\overleftrightarrow{r_3}(H)$ be the three-color directed Ramsey number, and let an acyclic digraph have maximum degree at most $\Delta$. Three-color directed-Ramsey conjecture. There is an absolute constant $\Delta$ and an infinite sequence $\{H_n\}$ of $n$-vertex acyclic digraphs, each with maximum degree at most $\Delta$, such that

$$\overleftrightarrow{r_3}(H_n)\geq n^{\omega(1)}.$$

The paper notes super-polynomial constructions for some other numbers of colors but identifies the three-color case as an unresolved intermediate case.