Proficiency conjecture for finite simple groups and their universal covers

Let $S$ be a finite simple group and let $\tilde{S}$ be its universal cover. A finite group is proficient when it has the relevant profinite presentation property discussed in the source. Proficiency conjecture. Every finite simple group $S$ and its universal cover $\tilde{S}$ are proficient. This question concerns whether all finite simple groups and their universal covers attain the minimal profinite presentation deficiency; the source notes that proficiency of $\tilde{S}$ is equivalent to a profinite presentation with $2$ generators and $2$ relations, while the conjecture remains unresolved there.