On minimal factorizations of words as products of palindromes

Given a finite word u, we define its palindromic length |u|_{pal} to be the least number n such that u=v_1v_2... v_n with each v_i a palindrome. We address the following open question: Does there exist an infinite non ultimately periodic word w and a positive integer P such that |u|_{pal}P. In particular, the result holds for all the k-power-free words and for the Sierpinski word.