Recall that for a profinite group $G$ we define the subgroup rank to be $$\sup \{d(H): H \leq_c G\}$$ where $d(H)$ stands for the minimal cardinality of a set of topological generators of $H$.
Let $p$ be a prime number, $F$ a free pro-$p$ group of rank $\aleph_0$, and let $G_i \lhd_c F$ be such that the subgroup rank of $F/G_i$ is bounded by some $k \in \mathbb{N}$ for all $i \in I$. Is it possible that $$\bigcap_{i \in I} G_i = \{1\} ?$$
