The blowup monotonicity conjecture of Korándi, Roberts and Scott

Fix $k\geq 2$, and let $\delta$ be small enough. For any $\delta>\delta_0>0$ and sufficiently large $n$, consider every $C_{2k-1}$-free graph $G$ on $n$ vertices satisfying

$$\left(\frac14-\delta_0\right)n^2\geq e(G)\geq\left(\frac14-\delta\right)n^2.$$

Korándi–Roberts–Scott's conjecture. There is a $C_{2k+1}$-blowup $G^*$ satisfying

$$e(G^*)\geq e(G)\qquad\text{and}\qquad\gamma_2(G^*)\geq\gamma_2(G).$$

The conjecture predicts that, in this dense near-bipartite range, an appropriate blowup preserves or improves both the edge count and the parameter $\gamma_2$; the supplied source gives no resolution status.