Frequency-module conjecture for Jacobi matrices of Cantor-type measures

Let $K(\gamma)$ be the Cantor-type set determined by $\gamma$, and let $H_{\mu_{K(\gamma)}}$ have recurrence-coefficient sequence $(a_n)_{n=1}^\infty$. An almost periodic sequence has a frequency module, the module generated by its Fourier frequencies modulo $1$. Frequency-module conjecture. For any $\gamma$, the sequence $(a_n)_{n=1}^\infty$ for $H_{\mu_{K(\gamma)}}$ is asymptotically almost periodic, and the almost periodic limit has frequency module

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

modulo $1$. The proposed frequency module reflects the dyadic integrated-density-of-states values observed in the gaps of the Cantor construction; the assertion is based on numerical Fourier-spectrum evidence and remains open.