Torsion-free groups small-doubling abelianity conjecture

Let $G$ be a torsion-free group and let $S$ be a finite subset of $G$ satisfying

$$|S|\geq 3,$$

and $e\in S$, where $e$ is the identity element of $G$. Write $S^2=\{s_1s_2:s_1,s_2\in S\}$ and let $\langle S\rangle$ denote the subgroup generated by $S$.

Small-doubling abelianity conjecture. If

$$|S^2|\leq 3|S|-3,$$

then $\langle S\rangle$ is an abelian subgroup of $G$.

The source describes this as a further conjecture believed to be true. It would give a general abelianity criterion for sufficiently small product sets in torsion-free groups, beyond the special decompositions into disjoint abelian sets.