Big ends eventually periodic conjecture

Let $\mathbb T_q$ be the tautological tree, whose ends determine sequences of $ell$-values and associated $F$-sequences. An end is big when

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

and it is of type $S$ when its associated $F$-sequence satisfies

$$F(m+1)=F(m)+1$$

for all sufficiently large $m$. Big ends eventually periodic conjecture. Every big end is of type $S$. The conjecture proposes that the summability condition defining big ends forces eventual type-$S$ behavior of the associated combinatorial sequence; its status is not resolved in the supplied source.