Brualdi–Mena conjecture on components of graphs from standard pairings

Let $(H,\Phi)$ be a standard pairing, where $H$ is a finite connected regular graph and $G(\Phi)^*$ denotes the graph associated with the pairing after removing singular components. Brualdi–Mena conjecture. If $H$ is isomorphic to a component of $G(\Phi)^*$, then $H$ is isomorphic to all components of $G(\Phi)^*$. This conjecture concerns the structure of graph components arising from standard pairings; the source states it as a conjecture in the context of graphs, and no resolution is supplied here.