Frid's logarithmic palindromic-length conjecture for power-free words

Let $w$ be an infinite word, and let $\operatorname{PPL}_w(n)$ denote the palindromic length of the prefix of $w$ of length $n$. Frid's conjecture. If $w$ is $k$-power-free for some positive integer $k$, then

$$\limsup_{n\rightarrow\infty}\frac{\operatorname{PPL}_w(n)}{\ln n}>0.$$

This predicts logarithmic growth, along an infinite subsequence of prefix lengths, for the palindromic length of prefixes of every power-free infinite word.