In this paper, we characterize all connected graphs for which the sum of two largest eigenvalues is less than $4$. As an application, we prove that the path graph minimizes this sum among all connected graphs of order $n\ge 467$, thereby solving a conjecture posed by Kumar, Liu, Monterde, Pragada, and Tait in "Maximum spectral sum of graphs (arXiv: 2604.00512v2)".