Balanced max-part conjecture for K_4-free graphs

Let $n$ be even and let $G$ be a $K_4$-free graph on $n$ vertices. A balanced $2$-partition is a partition $V(G)=A\cup B$ with $|A|=|B|=n/2$. Balanced max-part conjecture for K_4-free graphs. There exists a balanced $2$-partition such that each class spans at most $n^2/16$ edges:

$$\max\{e(A),e(B)\}\leq n^2/16.$$

The bound is sharp for the complete 3-partite graph with class sizes $n/2,n/4,n/4$, and the conjecture is proved in the source for 3-partite graphs but remains open in general.