Conference-graph lower-bound conjecture for complementary positive eigenvalue sums

Let $G$ be a graph on $n$ vertices, let $\overline G$ be its complement, and let $s^+(G)$ denote the sum of the squares of the positive adjacency eigenvalues. A conference graph is a strongly regular graph with parameters $(4t+1,2t,t-1,t)$. Conference-graph conjecture.

$$s^+(G)+s^+(\overline G)\ge\frac{(n-1)^2}{2}+\frac{(n-1)(n+1-2\sqrt n)}4=\frac{(n-1)(3n-1-2\sqrt n)}4,$$

with equality if and only if $G$ is a conference graph. The paper presents this as the proposed sharp lower bound; no general proof is supplied.