The bounded normal covering conjecture for finite simple groups

Let $c$ be a positive integer, and let $G$ be a finite non-abelian simple group. Write $\gamma(G)$ for its normal covering number. Bounded normal covering conjecture. There exists a function $f:\mathbb{N}\to\mathbb{N}$ such that if $\gamma(G)\leq c$, then one of the following holds:

1. $|G|\leq f(c)$;
2. $G$ is a finite simple group of Lie type having Lie rank at most $f(c)$;
3. $G=\mathrm{Sp}_n(q)$ with $q$ even.

Thus, for each fixed covering bound, the relevant finite simple groups should be small in number or constrained by bounded Lie rank, apart from the stated symplectic family. The source presents this as a clarification of what “small” means and does not give a resolution.