Unbounded palindromic length in aperiodic infinite words

For an infinite word $\mathbf{w}$, let the palindromic length $|u|_{\rm pal}$ of a finite factor or prefix $u$ be the least number $Q$ of palindromes whose concatenation is $u$. Unbounded palindromic-length conjecture. Every infinite word that is not ultimately periodic has factors—and even prefixes—with unbounded palindromic length. This is presented as an unresolved conjecture in general; the paper studies the question for Sturmian words and the relation between palindromes and the Sturmian numeration system.