Directed-cycle orientation-counting conjecture

Let $C_k^\circlearrowright$ denote the directed cycle of length $k$, and let $D(G(n,p),C_k^\circlearrowright)$ be the number of orientations of $G(n,p)$ containing no copy of $C_k^\circlearrowright$. Directed-cycle orientation-counting conjecture. If $p\gg n^{-1+1/(k-1)}$, then, with high probability,

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

This conjecture proposes a sharp form of the paper's bounds for directed cycles: the first term matches the known lower-bound scale, while the conjectured expression also accounts for the $n\log n$ contribution. The source does not provide a resolution.