Big components big ends conjecture for tautological laminations

Let $\mathcal S_q$ be the shift locus in a one-dimensional slice, and let $\mathbb T_q$ be the tautological tree whose ends correspond to components of the complement of $\mathcal S_q$. An end is big when its associated sequence of $ell$-values satisfies

$$\sum \frac{1}{\ell(n)}<\infty.$$

A component is said to have positive diameter when it is not a single point. Big components big ends conjecture. A component of the complement of $\mathcal S_q$ in a slice has positive diameter if and only if it corresponds to a big end of $\mathbb T_q$. This conjecture identifies the geometric distinction between point components and positive-diameter components with the combinatorial distinction between small and big ends; its status is not resolved in the supplied source.