The directed out-star inducibility conjecture

Let $\vec{S}_k$ be the directed out-star of order $k$, and let $\pi_{\vec{S}_k}(\emptyset)$ denote its inducibility among directed graphs. Define

$$\alpha_k=\max_{x\in[0,1]}\frac{kx(1-x)^{k-1}}{1-x^k}.$$

The directed out-star conjecture. For every $k\geq3$,

$$\pi_{\vec{S}_k}(\emptyset)=\alpha_k,$$

with the unique stable extremal configuration being a blow-up of $\vec{S}_2$ iterated inside part $1$, assigning a proportion $\alpha_k$ of the vertices to part $1$ at every iteration. The claim is known in the source for $k=3$ and $k=4$, but remains conjectural for general $k$.