Iterated blow-up conjecture for inducibility of directed paths

Let $\vec{P}_k$ be the directed path on $k$ vertices, and let $\vec{C}_{k+1}$ be the directed cycle on $k+1$ vertices. For an oriented graph $G$, write $d_{\vec{P}_k}(G)$ for the induced density of $\vec{P}_k$, and let $I(\vec{P}_k)$ be the limiting maximum of this density over all oriented graphs. An iterated balanced blow-up of $\vec{C}_{k+1}$ is obtained by repeatedly replacing each vertex by equally sized independent classes and orienting edges between classes according to the directed cycle.

Iterated blow-up conjecture. The number of induced copies of $\vec{P}_k$ over all oriented graphs on $n$ vertices is maximized by an iterated balanced blow-up of $\vec{C}_{k+1}$. Consequently,

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

The conjecture extends the announced result $I(\vec{P}_3)=2/5$ with extremal construction an iterated blow-up of $\vec{C}_4$. It is presented as an open generalization for longer oriented paths.