Falgas-Ravry–Vaughan conjecture for induced directed stars

Let $\vec{S}_k$ be the directed star on $k$ vertices, with one center and $k-1$ edges oriented away from it. For an integer $k\geq 3$, let $\pi_{\vec{S}_k}(\emptyset)$ denote the maximum induced density of $\vec{S}_k$ in a digraph. Define

$$\alpha_k=\max_{0\leq x\leq 1}\frac{kx(1-x)^{k-1}}{1-x^k}.$$

Assume the maximum is attained at $x=x_k$. Falgas-Ravry–Vaughan's conjecture.

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

and an extremal configuration is obtained by starting with parts $|A|=x_kn$ and $|B|=(1-x_k)n$, adding all edges oriented from $A$ to $B$, and iterating this construction inside $A$. The cases $k=3$ and $k=4$ are known by flag algebras, while the conjecture is open for general $k$, with computer-assisted methods becoming difficult as $k$ grows.