Dyadic minimum conjecture for recurrence coefficients of Cantor-type measures

Let $\gamma=(\gamma_k)_{k=1}^\infty$ be the parameter sequence defining $K(\gamma)$, and let $\mu_{K(\gamma)}$ be its associated measure with recurrence coefficients $(a_n)_{n=1}^\infty$. Dyadic minimum conjecture. For $\mu_{K(\gamma)}$,

$$\min_{i\in\{1,\dots,2^n\}}a_i=a_{2^n},$$

and, in particular,

$$\liminf_{s\rightarrow\infty}a_{2^s}=\liminf_{n\rightarrow\infty}a_n.$$

The claim is motivated by numerical experiments and is used in the paper to relate the behavior of the full recurrence-coefficient sequence to its dyadic subsequence; its resolution is not provided here.