Wall's conjecture on maximal proper subgroups of finite groups

For a finite group $G$, let $\max(G)$ denote the number of maximal proper subgroups of $G$. Wall's conjecture.

$$\max(G)\leq \lvert G\rvert.$$

This conjecture concerns a uniform bound on maximal proper subgroups. The source notes that a counterexample was found during a 2012 AIM workshop, so the conjecture is refuted.