Campos et al.'s Hilton condition conjecture for 3-clique graphs

Let $G$ be a 3-clique graph, and let Hilton's condition mean the condition for a simple graph with an even number of vertices and a universal vertex that

$$|E(\overline{G})|+\alpha(\overline{G})<\frac{|V(G)|}{2},$$

where $\alpha(\overline{G})$ is the cardinality of a maximum independent set of edges of $\overline{G}$. Campos et al.'s conjecture. The graph $G$ is type-II if and only if it satisfies Hilton's condition. The conjecture seeks to extend Hilton's characterization from the stated universal-vertex setting to 3-clique graphs; no resolution is given in the supplied text.