Let $F$ be a nonabelian finitely generated free profinite group, and let $H \leq F$ be a finitely generated closed subgroup. Must there be some open subgroup $H \leq U \leq F$, and a closed normal subgroup $N \lhd U$ such that $H \cap N = \{1\}$ and $HN = U$?

Note that the analogous statements are true if $F$ is free pro-$p$ group for any prime number $p$, if $F$ is a free group, if $F$ is a surface group and probably in other cases (which?).