The infinitude conjecture for cyclic groups with the weak acyclic matching property

Following Aliabadi, an abelian group $G$ has the weak acyclic matching property if, for every pair of finite nonempty subsets $A,B\subseteq G$ with $|A|=|B|$ and $A\cap(A+B)=\emptyset$, the set $A$ is acyclically matched to $B$. Here $\mathbb{Z}/n\mathbb{Z}$ denotes the cyclic group of order $n$. Infinitude conjecture. There are infinitely many $n$ for which $\mathbb{Z}/n\mathbb{Z}$ has the weak acyclic matching property. The conjecture asks whether this property occurs for infinitely many finite cyclic groups; its resolution is not indicated in the supplied source context.