Extremal-graph classification conjecture for triangle-free graphs

Let $n$ and $s$ be integers with $n/3<s\le n/2$, and let $\mathrm{ex}(n,s)$ denote the maximum number of edges in a triangle-free graph on $n$ vertices with independence number at most $s$. Let $g_k(n,s)$ be the functions defined in the preceding conjecture, and let $\mathscr{G}^n_k(s)$ and $\mathscr{H}^n_k(s)$ be the graph families described in the source. Extremal-graph classification conjecture. If $G$ has $\mathrm{ex}(n,s)$ edges, then $G$ is isomorphic to a graph in one of the families $\mathscr{G}^n_k(s)$ or $\mathscr{H}^n_k(s)$, and

$$\mathrm{ex}(n,s)=\min_k g_k(n,s).$$

The source proves the classification in a neighbourhood to the right of each critical ratio, but strongly suspects that it holds throughout the full range.