Breuer–Guralnick–Kantor Hamiltonian generating-graph conjecture

Let $G$ be a finite group with $|G|\geq 4$, and let $\Gamma(G)$ be its generating graph: the vertices are the non-identity elements, and $x$ and $y$ are adjacent exactly when $G=\langle x,y\rangle$. Breuer–Guralnick–Kantor's Hamiltonicity conjecture.

$$\Gamma(G)\text{ contains a Hamiltonian cycle}\quad\Longleftrightarrow\quad G/N\text{ is cyclic for every nontrivial normal subgroup }N\text{ of }G.$$

The cyclic-quotient condition is necessary, and the source reports the conjecture as proved for soluble groups and for groups ${\rm L}_2(q)$, with substantial progress for alternating groups; the general assertion remains unresolved.