The Nagy conjecture on stability for saturated graphs

Let $r\ge 2$ be an integer and let $F$ be a graph with chromatic number $r+1$. An $F$-saturated graph is an $F$-free graph to which adding any missing edge creates a copy of $F$. Let $t_r(n)$ denote the number of edges in the Turán graph with $r$ parts.

Nagy's conjecture. Every $F$-saturated graph $G$ on $n$ vertices with $t_r(n)-o(n^{\frac{r+1}{r}})$ edges contains a complete $r$-partite subgraph on $(1-o(1))n$ vertices.

This conjecture generalizes the corresponding stability results for saturated complete graphs and odd cycles. The exponent term cannot be improved in the clique case, and for $r=2$ it cannot be improved for any graph $F$ because every 3-chromatic graph contains an odd cycle. The conjecture is open in the stated generality.