The linear conjugacy-class bound for finite groups

Let $G$ be a finite group, let $k(G)$ denote its number of conjugacy classes, and let $|G|$ denote its order. Linear conjugacy-class bound. There exists a constant $C>0$ such that, for every finite group $G$,

$$k(G)\geq C\log_2|G|.$$

This conjecture asks for a universal logarithmic lower bound on the number of conjugacy classes in terms of the group order. It is presented as a central open problem motivating the paper's sharper result for alternating groups.