Extremal construction conjecture for generalized Ramsey–Turán numbers

Let $t>s\ge 3$. For $t\le 2s-1$, let $\mathcal{H}(n,s,t)$ be the family of graphs from the stated construction: partition the vertex set into $V_1\cup\cdots\cup V_s$, use an extremal $K_{t-s}$-free graph on $[s]$ to determine which pairs of classes are complete bipartite and which receive a Bollobás–Erdős graph, and put a triangle-free graph with sublinear independence number inside each class corresponding to a vertex of degree $s-1$. For $t\ge 2s$, let $\mathcal{H}(n,k)$ denote the family from Construction~. Extremal construction conjecture. One of the extremal graphs for ${\mathrm{RT}}(K_s,K_t,o(n))$ lies in $\mathcal{H}(n,s,t)$ when $t\le 2s-1$, and lies in $\mathcal{H}(n,k)$ with $k=t$ when $t\ge 2s$. The conjecture proposes a structural description of extremal graphs for generalized Ramsey–Turán problems; the source provides no resolution status in the supplied text.