Super-polynomial one-color oriented Ramsey growth for bounded-degree acyclic digraphs

For $k,\Delta\geq 1$ and $n>\Delta$, let $H_{k,n,\Delta}$ be an acyclic digraph with $n$ vertices and maximum degree $\Delta$ maximizing $\ovar{r_k}(H)$. Super-polynomial growth conjecture. There is an absolute constant $\Delta$ such that

$$\ovar{r_1}(H_{1,n,\Delta})\geq n^{\omega(1)}.$$

The paper establishes polynomial lower bounds and super-polynomial lower bounds for at least two colors, but leaves this one-color case open.