Prefix palindromic length conjecture for non-ultimately periodic words

Let $u=u[1]u[2]\cdots$ be an infinite word. For $n\in\mathbb N$, let $PPL_u(n)$ denote the minimum number of concatenated palindromes needed to express the prefix of $u$ of length $n$. Let $SP_u(k)$ denote the length of the shortest prefix of $u$ having palindromic length $k$, with $SP_u(k)=\infty$ if no such prefix exists. Prefix palindromic length conjecture. For every non-ultimately periodic word $u$, the function $PPL_u(n)$ is unbounded, equivalently,

$$SP_u(k)<\infty\quad\text{for every }k\in\mathbb N.$$

The conjecture was formulated by the author with Puzynina and Zamboni; it is known for $p$-power-free words and more generally for words satisfying a $(p,l)$-condition, and it has also been proved for all Sturmian words. It remains unresolved for some non-ultimately periodic words.