Let $G$ be a finite group. We define $r(G)$ to be the smallest number of relations possible in a presentation of $G$ with the minimal number of generators. If $G$ is a $p$-group, we can also consider "pro-$p$ presentations" of $G$ (using the free objects in the category of pro-$p$ groups); we write $r_p(G)$ for the smallest number of relations possible in a pro-$p$ presentation with the minimal number of generators.
Does $r(G) = r_p(G)$?