Algebraic lower-bound conjecture for the complement connectivity pair

Let $G$ be a graph on $n$ vertices, with complement $G^c$, and write

$$x=\lambda_2(G),\qquad y=\lambda_2(G^c).$$

Assume that $x<1$ and $y<1$.

Algebraic lower-bound conjecture. The pair $(x,y)$ satisfies

$$xy(2-xy)\geq n(1-x)(1-y)(n-2-x-y),$$

with equality only when $G$ belongs to one of the three graph families enumerated earlier in the paper. This empirically observed inequality is presented as a strengthening of the symmetric Laplacian Spread Conjecture; its status is unresolved in the source.