The spread and generating graph equivalence conjecture

Let $G$ be a finite group with $|G|\geqslant 4$, and let $\Gamma(G)$ be its generating graph. The spread of $G$ is the largest integer $k$ such that, for every set of $k$ nonidentity elements, there is an element that together with each member of the set generates $G$. Spread and generating graph equivalence conjecture. The following are equivalent:

$$\begin{array}{ll} \text{(i)}&G\text{ has spread }1;\\ \text{(ii)}&G\text{ has spread }2;\\ \text{(iii)}&\Gamma(G)\text{ has no isolated vertices};\\ \text{(iv)}&\Gamma(G)\text{ is connected};\\ \text{(v)}&\Gamma(G)\text{ is connected with diameter at most }2;\\ \text{(vi)}&\Gamma(G)\text{ contains a Hamiltonian cycle};\\ \text{(vii)}&G/N\text{ is cyclic for every nontrivial normal subgroup }N. \end{array}$$

The conjecture combines and strengthens conjectures of Breuer, Guralnick and Kantor and of Breuer, Guralnick, Lucchini, Maróti and others. Several implications are immediate, and the equivalence of (vi) and (vii) is known for soluble groups; the full equivalence remains open.