The componentwise inertia sum-of-squares conjecture

Let $G$ be a simple undirected graph with $n$ vertices and $\kappa$ connected components. Let $\mu_1\geq\cdots\geq\mu_n$ be the eigenvalues of its adjacency matrix, and let $\pi$ and $\nu$ be the numbers of positive and negative eigenvalues, counted with multiplicity. Define

$$s^+=\sum_{i=1}^{\pi}\mu_i^2,\qquad s^-=\sum_{i=n-\nu+1}^{n}\mu_i^2.$$

Componentwise inertia sum-of-squares conjecture. One has

$$\min\{s^-,s^+\}\geq n-\kappa.$$

For a connected graph, this specializes to the conjectured lower bound $\min\{s^-,s^+\}\geq n-1$, and either one-sided inequality implies the corresponding Hong-type bound. The supplied source gives no evidence of a resolution for the general disconnected-graph formulation.