Balogh–Clemen–Lidický conjecture on the balanced bipartite distance of $K_4$-free graphs

Let $G$ be a $K_4$-free graph on $n$ vertices. The Balogh–Clemen–Lidický conjecture. $G$ can be made balanced bipartite by removing at most

$$\frac{n^2}{9}$$

edges. This conjecture asks whether the sharp bipartite-distance bound for $K_4$-free graphs matches the bound suggested by the complete tripartite graph. The surrounding text identifies it as a conjecture of Balogh, Clemen, and Lidický; no resolution is given here.