Conjecture on controllable graphs with complete auxiliary graphs

For a graph $G$, let $\Omega(G)$ denote the auxiliary graph used in the paper's generalized spectral characterization, and call $G$ controllable when it satisfies the controllability condition defined there. Controllable-graph conjecture. Almost all controllable graphs $G$ have complete graphs as $\Omega(G)$. This conjecture is motivated by the paper's simulations: among the tested graphs, the vast majority of those guaranteed to be determined by their generalized spectra had $\Omega(G)=K_n$. Its general validity remains open.