Hamidoune's atom cardinality conjecture for torsion-free groups

Let $G$ be a torsion-free group, let $C$ be a finite subset of $G$ containing the identity element, and let $n$ be a positive integer. For a finite subset $X\subseteq G$, write $\partial_C(X)=XC\setminus X$, let $\kappa_n(C)$ be the minimum of $|\partial_C(X)|$ over finite $X\subseteq G$ with $|X|\geq n$, and let an $n$-atom of $C$ be an $n$-critical set of minimal cardinality; denote its cardinality by $\alpha_n(C)$. Hamidoune's conjecture. Every $n$-atom of $C$ has cardinality $n$, that is, $|\alpha_n(C)|=n$. The conjecture is known for unique product groups and for $n=1$, but remains open in general; attempts to prove it even for $n=2$ have been unsuccessful.