Two disjoint abelian sets small-doubling conjecture

Let $G$ be a torsion-free group, and let $S$ be a nonempty finite subset of $G$ written as

$$S=A\cup B,$$

where $A$ and $B$ are disjoint abelian sets. Write $S^2=\{s_1s_2:s_1,s_2\in S\}$ and let $\langle S\rangle$ denote the subgroup generated by $S$.

Two-set small-doubling conjecture. If $|S^2|\leq 3k-3$, then $\langle S\rangle$ is abelian.

This conjecture is proposed as a possible extension of the preceding results on small doubling in torsion-free groups. Its status is not resolved in the source.