Economical separability in free groups

Consider the rank $n$ free group $F_n$ with basis $X$. Bogopol'skii conjectured in [1, Problem 15.35] that each element $w\in F_n$ of length $|w|\ge2$ with respect to $X$ can be separated by a subgroup $H\le F_n$ of index at most $\le C\log|w|$ with some constant $C$. We prove this conjecture for all $w$ outside the commutant of $F_n$, as well as the separability by a subgroup of index at most $\frac{|w|}2+2$ in general.