On the almost-palindromic width of free groups

We answer a question of Bardakov (Kourovka Notebook, Problem 19.8) which asks for the existence of a pair of natural numbers $(c,m)$ with the property that every element in the free group on the two-element set $\{a,b\}$ can be represented as a concatenation of $c$, or fewer, $m$-almost-palindromes in letters $a^{\pm 1},b^{\pm 1}$. Here, an $m$-almost-palindrome is a word which can be obtained from a palindrome by changing at most $m$ letters. We show that no such pair $(c,m)$ exists. In fact, we show that the analogous result holds for all non-abelian free groups.