Quantitative growth conjecture for multicolor oriented Ramsey numbers

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 $\ovar{r_k}(H)$. Quantitative growth conjecture. There exist $\Delta\geq 3$ and constants $c_k=\omega_k(1)$ and $C_k=2^{o(k)}$ such that

$$\Omega(\log^{c_k} n)\leq \log \ovar{r_k}(H_{k,n,\Delta})\leq O(\log^{C_k} n).$$

This is proposed as a quantitative refinement of the open question of determining the logarithmic order of these Ramsey numbers; the source gives no resolution evidence.