The general abelian-structure conjecture for approximate groups

Let $G$ be a group and let $A\subset G$ be a finite symmetric set satisfying

$$|A^3|\leq K|A|.$$

A subset is commuting if all of its elements commute pairwise. The general abelian-structure conjecture. There exists an absolute constant $c>0$ such that there is a commuting set $T\subset A^4$ satisfying

$$|T|\geq\exp\left(\Omega\left(\frac{\log^c|A|}{\log 2K}\right)\right).$$

The conjecture would remove the hypothesis excluding alternating-group subquotients from the paper's abelian-structure theorem. The paper proves weaker versions in torsion-free groups and when $|A|$ is sufficiently large in terms of $K$, while the full statement remains open.