A question about bases of free groups

Question

Let $F_n$ be a free group of finite rank $n$ and let $\gamma_m(F)$ denote the $m$-th term of the lower central series of $F$ where $m \ge 1$ is a natural number. Suppose that $x,y$ are primitive elements of $F$ that are equivalent modulo $\gamma_m(F)$: $$ x \equiv y\, (\mathrm{mod}\, \gamma_m(F)). $$ Can then both $\{x\}$ and $\{y\}$ be extended to bases $\{x,x_2,\ldots,x_n\}$ and $\{y,y_2,\ldots,y_n\}$ of $F_n$ so that $$ x_k \equiv y_k\, (\mathrm{mod}\, \gamma_m(F)) $$ for all $k=2,\ldots,n?$ Apparently, the statement is true for $n=2,$ but what about greater $n?$