*(converted from the comments)* No, $F_2\times F_2$ is a counterexample, where $F_2$ is free on 2 generators.

Recall that a group is LPF if the profinite closure of every f.g. subgroup of infinite index has infinite index. This fails if there is a profinitely dense f.g. subgroup.

[Remark: your property appeared in the 3-manifold literature as "$G$ has the engulfing property". It is an elementary remark (see page 10-11 here, where Property LPF is introduced) that a group has Property LPF iff each of its finite index subgroups has the engulfing property.]

Now a way to get profinitely dense f.g. subgroups in $F_2\times F_2$ is as follows. Consider an aperiodic (= with no nontrivial finite quotient) infinite finitely presented group $P$ on 2 generators (or with $n$ generators, but then work with $F_n\times F_n$), and fix an epimorphism $f:F_2\to P$. Then the fibre product $$F_2\times_P F_2=\{(x,y)\in F_2\times F_2:f(x)=f(y)\}$$ is profinitely dense in $F_2\times F_2$ (because $P$ is aperiodic) and finitely presented (because $P$ is finitely presented) and has infinite index (because $P$ is infinite).

presentedsubgroup of a residually free group is separable. As well as my result about limit groups, the extra ingredient is a theorem of Bridson--Howie--Miller--Short, and the answer to your question might be in their work. $\endgroup$