19.8 (2018)

A word in an alphabet $A = \{a_1, a_2, \dots, a_n\}$ is called a palindrome if it reads the same from left to right and from right to left. Let $k$ be a non-negative integer. A word in the alphabet $A$ is called an almost $k$-palindrome if it can be transformed into a palindrome by changing $\leqslant k$ letters in it. (So an almost 0-palindrome is a palindrome.) Let elements of a free group $F_2 = \langle x, y \rangle$ be represented as words in the alphabet $\{x^{\pm 1}, y^{\pm 1}\}$. Do there exist positive integers $m$ and $c$ such that every element in $F_2$ is a product of $\leqslant c$ almost $m$-palindromes?

It is known that for $m = 0$ there is no such a number $c$.