Negative square energy versus graph surplus

For a graph $G$, let $\operatorname{surp}(G)$ denote its surplus, and let $s^-(G)$ denote its negative square energy. Negative surplus exponent conjecture.

$$s^-(G)=\Omega(\operatorname{surp}(G)^{6/7}).$$

The paper constructs examples showing that the corresponding positive-energy relation does not transfer directly to negative square energy and conjectures that the exponent $6/7$ is the worst-case behavior. The general bound remains open.