Strongly connected tournament orientation-counting conjecture

Let $\vec{H}$ be a strongly connected tournament with $k:=v(H)\geqslant 4$, and let $p\gg n^{-2/(k+1)}$. Tournament orientation-counting conjecture. Then, with high probability,

$$\log D(G(n,p),\vec{H})=\widetilde{\Theta}\left(\frac{n}{p^{(k-1)/2}}\right).$$

Estimating the number of $\vec{H}$-free orientations remains open for strongly connected tournaments with at least four vertices; a more general version of this conjecture appeared in earlier work.