Directed-path inducibility conjecture for transitive-tournament-free oriented graphs

Let $\vec{P}_k$ be the directed path on $k$ vertices, let $\vec{C}_{k+1}$ be the directed cycle on $k+1$ vertices, and let $\vec{T}_3$ be the transitive tournament on three vertices. Let $\mathcal{\vec T}$ be the family of $\vec{T}_3$-free oriented graphs, and write $I(\vec{P}_k,\mathcal{\vec T})$ for the limiting maximum induced density of $\vec{P}_k$ in this family.

Transitive-tournament-free inducibility conjecture. The number of induced copies of $\vec{P}_k$ over all $\vec{T}_3$-free oriented graphs on $n$ vertices is maximized by a balanced blow-up of $\vec{C}_{k+1}$. Consequently,

$$I(\vec{P}_k,\mathcal{\vec T})=\frac{k!}{(k+1)^{k-1}}.$$

This is a restricted version of the iterated blow-up conjecture, motivated by the simpler structure of $\vec{T}_3$-free oriented graphs; its resolution is not supplied in the source.