Conjectured formula for the uniformly dense Turán density of stars

For each integer $k$ with $4\le k\le 10$, let $S_k$ denote the star $3$-graph considered in the paper, and let $\pi_{\mathrm{U}}(S_k)$ be its uniformly dense Turán density. Conjecture for stars.

$$\pi_{\mathrm{U}}(S_k)=\frac{k^2-5k+7}{(k-1)^2}\quad\text{for all }4\le k\le 10.$$

The paper proves the same formula for $k\ge 11$ and, for the related notion represented by $\pi_{\mathrm{V}}$, for $k\ge 5$; the uniformly dense cases $4\le k\le 10$ remain conjectural.