The containment conjecture for the global-breadth class

Let $\mathcal{H}$ be the class of finite groups satisfying

$$|G|\leq \mathbf{B}(G)(\mathbf{B}(G)+1),$$

and let Theorem

denote the paper's classification of groups under discussion. **Containment conjecture.** The class $\mathcal{H}$ might contain all groups in Theorem

. It might also contain simple and nonsimple groups that do not necessarily possess a nontrivial partition. The problem is explicitly presented as open in the paper; the broader possibility that $\mathcal{H}$ contains groups without nontrivial partitions is likewise unresolved.