The path as the unique minimum spectral-sum graph

Let $n$ denote the order of a connected graph, and let the spectral sum be $\lambda_1(G)+\lambda_2(G)$, where $\lambda_1(G)$ and $\lambda_2(G)$ are the largest and second-largest adjacency eigenvalues. The path-minimization conjecture. For sufficiently large $n$, the path uniquely minimizes the spectral sum among all connected graphs of order $n$. The minimization of spectral sum remains open, and this conjecture proposes the extremal graph for the connected case.