Let $F_n$ be a free profinite group of finite rank $n$ and let $V_k$ denote the intersection of all open subgroups of $F_n$ of rank at most $k$ ($k \in \mathbb{N}$).

My questions are:

can I explicitly compute the index of $V_k$ in $F_n$ for all $k \in \mathbb{N}$?

Or if not, maybe for some special $k$, e.g. $k = p$, $p$ prim?

I came to that question when I looked up the proof that the automorphism group of a finitely generated free profinite group is profinite (see e.g. Proposition 4.4.3 in Profinite Groups by Ribes-Zalesskii).

Thanks for any help