On palindromic factorization of words

Given a finite word u, we define its palindromic length |u|pal to be the least number n such that u=v1v2…vn with each vi a palindrome. We address the following open question: let P be a positive integer and w an infinite word such that |u|pal⩽P for every factor u of w. Must w be ultimately periodic? We give a partial answer to this question by proving that for each positive integer k, the word w must contain a k-power, i.e., a factor of the form uk. In particular, w cannot be a fixed point of a primitive morphism. We also prove more: for each pair of positive integers k and l, the word w must contain a position covered by at least l distinct k-powers. In particular, w cannot be a Sierpinski-like word.