The unistructurality conjecture for cluster algebras

Let $\mathcal A$ be a cluster algebra, with set of cluster variables $\mathcal X$. The algebra is unistructural if the set of cluster variables determines the cluster algebra structure, equivalently if there is a unique decomposition of $\mathcal X$ into clusters.

Unistructurality conjecture. Every cluster algebra is unistructural.

The paper proves unistructurality for cluster algebras of Dynkin type and rank $2$. The assertion is used to establish the cluster automorphism conjecture in the unistructural case, but remains open in general in the source.