The Bollobás–Nikiforov conjecture for $K_4$-free graphs

Let $G$ be a $K_4$-free graph with $m$ edges, and let

$$\lambda_1(G)\geq\lambda_2(G)\geq\cdots\geq\lambda_n(G)$$

be the adjacency eigenvalues of $G$. Bollobás–Nikiforov conjecture for $K_4$-free graphs. If $G\neq K_3$, then

$$\lambda_1^2(G)+\lambda_2^2(G)\leq \frac{4m}{3}.$$

This is the unresolved $K_4$-free specialization of the full conjecture. The source proves the bound for complete multipartite graphs and for sufficiently large dense $K_4$-free graphs, leaving cases with chromatic number at least four and $m=o(n^2)$ or small $n$ open.