Balanced 3-partition conjecture for K_4-free graphs

Let $n$ be divisible by $3$ and let $G$ be a $K_4$-free graph on $n$ vertices. A balanced $3$-partition divides $V(G)$ into three classes of size $n/3$; class-edges are edges whose endpoints lie in the same class. Balanced 3-partition conjecture for K_4-free graphs. There exists a balanced $3$-partition with at most

$$\frac{4}{81}n^2$$

class-edges. This is one of the paper's open partition problems for $K_4$-free graphs, and no general resolution is supplied.