The palindrome-to-factor complexity ratio conjecture for non-ultimately periodic words

Let ${\bf x}=(a_i)_{i\geq 0}$ be an infinite word over a finite alphabet $\Sigma$. Write $\rho_{\bf x}(n)$ for the number of distinct length-$n$ factors of ${\bf x}$, and let $\operatorname{Pal}_{\bf x}(n)$ be the number of distinct length-$n$ palindromes occurring in ${\bf x}$. Palindrome-to-factor complexity ratio conjecture. If ${\bf x}$ is not ultimately periodic, then

$$\lim_{n\to\infty}\frac{\operatorname{Pal}_{\bf x}(n)}{\rho_{\bf x}(n)}=0.$$

The conjecture proposes that, for every non-ultimately periodic infinite word, palindrome complexity is asymptotically negligible compared with factor complexity; the examples discussed include the Thue–Morse word and the binary concatenation word, for which this ratio tends to zero. Its general status is not specified in the source.