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 for each factor u of w? We give a partial answer to this question by proving that if an infinite word w satisfies the so-called (k,l)-condition for some k and l, then for each positive integer P there exists a factor u of w whose palindromic length |u|_{pal}>P. In particular, the result holds for all the k-power-free words and for the Sierpinski word.