Elphick–Farber–Goldberg–Wocjan positive and negative square-energy conjecture

Let $G$ be a finite simple undirected graph with $n$ vertices, adjacency eigenvalues $\lambda_1,\ldots,\lambda_n$, and positive and negative square energies

$$s^+(G):=\sum_{\lambda_i>0}\lambda_i^2,\qquad s^-(G):=\sum_{\lambda_i<0}\lambda_i^2.$$

Elphick–Farber–Goldberg–Wocjan conjecture. Every connected graph $G$ on $n$ vertices satisfies

$$\min\{s^+(G),s^-(G)\}\ge n-1.$$

The paper states that this conjecture is proved using a framework based on doubly nonnegative matrix relaxations of positive semidefinite matrix Hadamard squares.