The maximal semi-inducibility conjecture for the 4-vertex graph H_3

Let $H_3$ be the 4-vertex graph with blue edges $\{0,1\}$ and $\{1,2\}$ and red edge $\{2,3\}$. For $0\leqslant\gamma\leqslant\beta\leqslant 1$, define

$$p(\beta,\gamma):=\beta(1-\beta)(1-\gamma)(1-\beta)+(1-\beta)\beta\gamma\beta+\beta(\beta-\gamma)(1-\gamma)\gamma.$$

Let $\beta$ and $\gamma$ be the roots specified by

$$\beta^5-\frac{10}{3}\beta^4+\frac{2521}{576}\beta^3-\frac{407}{144}\beta^2+\frac{43}{48}\beta-\frac19=0$$

and

$$\gamma^4-\frac{37}{16}\gamma^3+\frac{57}{32}\gamma^2-\frac{9}{16}\gamma+\frac{1}{16}=0.$$

Semi-inducibility conjecture for $H_3$. The semi-induced density satisfies

$$\lambda_{H_3}=p(\beta,\gamma),$$

where the relevant roots are approximately $\beta=0.39829918$ and $\gamma=0.28158008$, giving $p(\beta,\gamma)\approx 0.150083407311578$.

This would round the flag-algebra lower bound for the only remaining unknown case in the table of 4-vertex graphs. The displayed construction produces the stated lower bound, and the numerical flag-algebra value appears to match it, but the authors have not been able to prove the exact value.