Sudakov's bipartisation conjecture for K_{r+1}-free graphs

For a graph $G$, let $D_2(G)$ be the minimum number of edges that must be deleted to make $G$ bipartite. Fix $r\geq3$, and let $G$ be an $n$-vertex $K_{r+1}$-free graph. Sudakov's bipartisation conjecture.

$$D_2(G)\leq\begin{cases}\frac{(r-1)^2}{4r^2}n^2,& r\text{ odd},\frac{r-2}{4r}n^2,& r\text{ even}.\end{cases}$$

The conjecture is known for $r=5$ by flag algebras, whereas the even-$r$ cases are described as more difficult and remain open.