Asymptotic almost periodicity conjecture for Widom–Hilbert factors

Let $K(\gamma)$ be the Cantor-type set associated with $\gamma$, let $\mu_{K(\gamma)}$ be its associated measure, and let $W_n^2(\mu_{K(\gamma)})$ be the corresponding Widom–Hilbert factors. Widom–Hilbert almost-periodicity conjecture.

$$\left(W_n^2\left(\mu_{K(\gamma)}\right)\right)_{n=1}^\infty$$

is asymptotically almost periodic if and only if $K(\gamma)$ is Parreau–Widom. If $K(\gamma)$ is Parreau–Widom, then the frequency module of the almost periodic limit includes the module generated by

$$\{m2^{-n}\}_{m,n\in\{N_0\}}$$

modulo $1$. The claim would connect asymptotic almost periodicity of Widom–Hilbert factors to Parreau–Widom geometry; the source presents it as a conjecture based on numerical spectra and leaves it open.