The digit-determining conjecture for non-strongly recurrent quadratic laminations

Let $\alpha\in\mathbb{T}$ be a parameter, and let $GCS_n(\alpha)$ denote the generalized cylinder set associated with the first $n+1$ letters of the itinerary $I(\alpha)$. Say that $\alpha$ is strongly recurrent according to the preceding definition, and call it pre-periodic when its associated itinerary is pre-periodic. Digit-determining conjecture. If $\alpha$ is not strongly recurrent and is not pre-periodic, then the statement of Proposition~ still holds: for every integer $L>0$, there exists an integer $L_\alpha>0$ such that

$$GCS_n(\alpha)\cap\{h^i(\alpha)\}_{i=1}^L=\emptyset\quad\text{for all }n\geq L_\alpha.$$

The conjecture removes the non-weakly-pre-periodic hypothesis from Proposition~. If true, the paper notes that all points on $\mathbb{T}$ except for a set of Hausdorff dimension zero would satisfy $CCE$; the claim remains open in the supplied text.