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±1,b±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.