An upper bound for the length of a Traveling Salesman path in the Heisenberg group

We show that a sufficient condition for a subset $E$ in the Heisenberg group (endowed with the Carnot-Carath'{e}odory metric) to be contained in a rectifiable curve is that it satisfies a modified analogue of Peter Jones’s geometric lemma. Our estimates improve on those of \cite{FFP}, by replacing the power $2$ of the Jones-$\beta$-number with any power $r<4$. This complements (in an open ended way) our work \cite{Li-Schul-beta-leq-length}, where we showed that such an estimate was necessary, but with $r=4$.