Let $F$ be a nonabelian finitely generated free pro-$p$ group, $H \leq_c F$ of infinite index. Denote by $\{F_n\}_{n \in \mathbb{N}}$ the lower central series of $F$, and set $r_n = [F : F_nH]$.
Is it poosible that for all $n \in \mathbb{N}$, $r_n < \infty$ ?
If this is possible, I would like to have a bound from below on the growth rate $r_n \rightarrow \infty$.
