This is an attempt to find a correct version of Do free profinite groups satisfy Howson's theorem?
Let $F$ be a free profinite group, and let $A,B \leq F$ be closed finitely generated subgroups which (topologically) generate $F$ (i.e. the closure of the subgroup generated by $A$ and $B$ is $F$).
Must $A \cap B$ be (topologically) finitely generated?
Some intuition comes from the fact that the previous counterexample from Do free profinite groups satisfy Howson's theorem? does not work here. Additional motivation is the fact that in the classical Howson's theorem (where $F$) is an abstract free group the additional condition $\langle A \cup B \rangle = F$ holds withou loss of generality since a subgroup of a free group is free. Hence, the same holds in the pro-$p$ case (for any prime $p$) where there exists an analogous theorem.