Elphick–Tang–Zhang square-energy conjecture for connected graphs

Let $G$ be a connected graph on $n$ vertices, and let $\mathcal{E}_2^+(G)$ and $\mathcal{E}_2^-(G)$ denote its positive and negative $2$-energies, respectively. Elphick–Tang–Zhang's conjecture.

$$\min\{\mathcal{E}_2^+(G),\mathcal{E}_2^-(G)\}\geq n-1.$$

The conjecture has been verified for several classes of graphs, but remains open in general; the best-known bound stated here is $\min\{\mathcal{E}_2^+(G),\mathcal{E}_2^-(G)\}\geq 3n/4$.