Positive-measure and recurrence-coefficient conjecture for Cantor-type sets

Let $\gamma=(\gamma_k)_{k=1}^\infty$ be the parameter sequence defining $K(\gamma)$, set $\varepsilon_k:=1-4\gamma_k$ for $k\in\mathbb{N}$, and let $(a_n)_{n=1}^\infty$ be the recurrence coefficients for $\mu_{K(\gamma)}$. Positive-measure conjecture.

$$K(\gamma)\text{ has positive Lebesgue measure} \quad\Longleftrightarrow\quad \sum_{s=1}^\infty\sqrt{\varepsilon_s}<\infty \quad\Longleftrightarrow\quad \liminf_{n\rightarrow\infty}a_n>0.$$

The conjecture would characterize positive Lebesgue measure simultaneously through the construction parameters and the asymptotics of the Jacobi recurrence coefficients. The paper presents it as an inference from numerical experiments and does not establish the equivalences.