Let $G$ be a finitely generated residually finite group and $\hat G$ its profinite completion. Then for all $g\in \hat G$ we have $gGg^{-1}\leq \hat G$ is dense.

Suppose that $H\leq \hat G$ is a closed subgroup such that $H\cap G$ is dense in $H$.

Can we say something in general on the density of $H\cap gGg^{-1}$ in $H$ ? Or if we assume that $G$ is a $p$-group (and so $\hat G$ a pro-$p$ group) ?

N.B. if $A\leq \hat G$ is of finite index, then $A\cap G$ is dense in $A$, hence the question is interesting only for infinite index subgroups.