Erdős–Hajnal conjecture for the extremal function $h_1^{(3)}$

Let $\mathcal{F}$ be the minimal family of 3-uniform hypergraphs generated from the empty hypergraphs on $1$ and $2$ vertices and an edge by the operation $H(G,v)$. Let $g_1^{(3)}(s)$ be the maximum number of edges in a member of $\mathcal{F}$ with $s$ vertices, and let $h_1^{(3)}(s)$ be the transition value from the generalized Ramsey-function conjecture. Erdős–Hajnal conjecture. For every positive integer $s$,

$$h_1^{(3)}(s)=g_1^{(3)}(s)+1.$$

The source notes the already established inequality $h_1^{(3)}(s)>g_1^{(3)}(s)$ and conjectures that this lower bound is tight.