Erdős–Hajnal equality conjecture for the functions $F_1$ and $g_1^{(3)}$

For a positive integer $s$, let $F_1(s)$ be the maximum number of triples $(a,b,c)$ with $1\leq a<b<c\leq s$ satisfying a prescribed three-coloring pattern on the edges $(a,b)$, $(b,c)$, and $(a,c)$, maximized over all edge-colorings of the complete graph on $[s]$ with colors $I$, $II$, and $III$. Let $g_1^{(3)}(s)$ be the maximum number of edges in a member of the recursively defined family $\mathcal{F}$ of 3-uniform hypergraphs on $s$ vertices. Erdős–Hajnal equality conjecture. For every positive integer $s$,

$$F_1(s)=g_1^{(3)}(s).$$

The source explains that this equality would imply $h_1^{(3)}(s)=F_1(s)+1=g_1^{(3)}(s)+1$, and hence the preceding Erdős–Hajnal conjecture; no resolution is given.