Three disjoint abelian sets strengthened small-doubling conjecture

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

$$S=A\cup B\cup C,$$

where $A$, $B$, and $C$ are pairwise 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 $S$.

Three-set small-doubling conjecture. If $|S^2|\leq 3k-4$, then $\langle S\rangle$ is abelian.

The source gives an example showing that the analogous conclusion need not hold for three abelian sets when $|S^2|=3|S|-3$. The conjecture proposes the stricter threshold $3k-4$, but no resolution is supplied.