The projective-rank lower bound for graph energy

Let $G$ be an $n$-vertex graph, and let $\xi_f(G)$ denote the projective rank of $G$. Projective-rank energy conjecture.

$$\frac{1}{2}\mathcal{E}(G) \geq n-\xi_f(\overline{G}).$$

This is proposed as a weakening of Fajtlowicz's graph energy conjecture and is motivated by the paper's semidefinite-programming lower bound involving the fractional chromatic number. The source presents it as a future research direction and gives no resolution.