Frid's Fibonacci-word palindromic-length lower-bound conjecture

Let $\boldsymbol{f}$ be the Fibonacci word, namely the fixed point of the morphism $0\mapsto 01$, $1\mapsto 0$. Let $\tau$ denote the golden ratio. Frid's Fibonacci-word lower-bound conjecture. The palindromic-length function of the Fibonacci word should satisfy

$$\limsup_{n\to\infty}\frac{{\mathrm{PL}_{\boldsymbol{f}}}(n)}{\ln n}\geq\frac{1}{3\ln\tau}.$$

This is presented as an unproved conjecture about the lower growth of palindromic length; even a lower bound of this kind was not known for the Fibonacci word in the paper.