Given a countable family of (non-abelian) torsion groups $G_n$ (i.e. each element has a finite order) in an inverse system $G_1\leftarrow G_2\leftarrow\dots G_n\leftarrow\dots$, where the maps are assumed to be surjective and given that the inverse limit $G=\lim_\leftarrow G_n$ is torsion, can we derive that the set of these groups has some uniform bound $L$ on their orders, i.e. that, for all but finitely many n, $G_n^L=1$.
I do not know the answer but I believe that it is a "yes".