The color-critical-edge stability conjecture for saturated graphs

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

Color-critical-edge stability 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 $G'$ on $(1-o(1))n$ vertices.

This is proposed as a weaker version of the general conjecture, restricting $F$ to graphs with a color-critical edge. The paper presents it as an open proposed version; the full conjecture is proved only for certain 3-chromatic graphs under additional hypotheses.