The connectivity conjecture for the generating graph of finite groups

Let $G$ be a finite group, and let $\bar\Delta(G)$ denote its generating graph, whose vertices represent the elements of $G$ and whose edges join pairs that generate $G$. Generating-graph connectivity conjecture. The graph $\bar\Delta(G)$ is connected for every finite group $G$. This conjecture concerns how generating pairs are distributed among the elements of finite groups. Connectivity is known for several classes, including soluble and characteristically simple groups, but the assertion for all finite groups remains open.