The weak-normal covering conjecture for finite simple groups

Let $G$ be a finite non-abelian simple group. Write $\gamma(G)$ for its normal covering number and $\gamma_w(G)$ for its weak normal covering number, and let $m\in\mathbb{N}$. Weak-normal covering conjecture. There exists a function $f:\mathbb{N}\to\mathbb{N}$ such that, if $\gamma_w(G)=m$, then either

$$\gamma(G)=m,$$

or $|G|\leq f(m)$, or $G\cong\mathrm{P}\Omega_n^+(q)$. The conjecture predicts that discrepancies between the two covering numbers are confined to bounded-order groups and the indicated orthogonal groups; the latter are expected to provide infinitely many discrepancies because of graph automorphisms.