Strengthened lower bound for the sum of the two smallest graph eigenvalues

Let $G$ be a graph of order $n$, and let $\lambda_{n-1}$ and $\lambda_n$ denote its two smallest eigenvalues. Strengthened eigenvalue conjecture. One has

$$\lambda_{n-1}+\lambda_n\geq -\frac{2n}{3}.$$

This strengthens the original conjectured bound $c_3=\frac{1}{3}$ and was verified for all graphs of order at most $9$; the general case remains open.