Erdős–Hajnal–Simonovits–Sós–Szemerédi periodic structure conjecture for Ramsey–Turán extremal graphs

Let $G$ be an asymptotically extremal graph for the Ramsey–Turán density $\varrho_p(q)$, where the $p$-independence number is the largest size of a vertex set inducing a $K_p$-free graph. Let $q=pt+r+2$, where $t\in\mathbb{N}$ and $r\in\mathbb{Z}_p$. Then there exists a partition

$$V(G)=V_0\cup V_1\cup\cdots\cup V_t$$

such that $e(G[V_i])=o(n^2)$ for every $0\leq i\leq t$, $d_G(V_0,V_1)=\frac{r+1}{p}-o(1)$ with degrees in $G[V_0,V_1]$ differing by $o(n)$, and $d_G(V_i,V_j)=1-o(1)$ for every pair $\{i,j\}\neq\{0,1\}$. Periodic structure conjecture. The asymptotically extremal graph has this structure; in particular,

$$\varrho_p(q)=\varrho_p^\star(q):=\frac{(t-1)(2p-r-1)+r+1}{t(2p-r-1)+r+1}.$$

This conjecture predicts a periodic structure for asymptotically extremal graphs in Ramsey–Turán problems, extending the fully determined case $p=2$. The cited work of Erdős, Hajnal, Simonovits, Sós and Szemerédi is the source of the conjecture; the material provided does not state whether it has been resolved.