Conjectured value of the uniformly dense Turán density of the four-vertex star

Let $S_4$ be the four-vertex star $3$-graph, and let $\pi_{\mathrm{V}}(S_4)$ denote the corresponding Turán-density parameter. Conjecture for the four-vertex star.

$$\pi_{\mathrm{V}}(S_4)=\frac{1}{3}.$$

This is the second conjecture announced in the concluding remarks, following the paper's results for the other parameter ranges; no resolution is supplied in the given text.