Freiman's small-doubling conjecture for finite subsets of torsion-free groups

Let $A$ be a finite subset of a torsion-free group, with $|A|\geq 4$. A progression is a finite geometric progression in the group. Freiman's conjecture. If

$$|A^2|\leq 3|A|-4,$$

then $A$ is covered by a progression of length at most $2|A|-3$. This conjecture extends the $(3k-4)$-theorem from abelian additive combinatorics to torsion-free groups; the source gives no resolution of it.