Keevash–Sudakov conjecture on monochromatic copies and Turán numbers

Let $H$ be a graph, and let $f(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of $H$ in a $2$-edge-coloring of the complete graph $K_n$. Keevash–Sudakov conjecture. For any graph $H$ and sufficiently large $n$,

$$f(n,H)=ex(n,H).$$

The paper notes that its main theorem provides counterexamples with chromatic number $3$, so the conjecture is false in general.