Inequality conjecture between cluster and upper cluster algebras

Let $\Sigma_{g,n}$ be a surface of genus $g$ with $n$ punctures, and let $\mathcal{A}(\Sigma_{g,n})$ and $\mathcal{U}(\Sigma_{g,n})$ denote its cluster and upper cluster algebras, respectively. Cluster–upper algebra inequality conjecture. For any $g,n\geq 1$,

$$\mathcal{A}(\Sigma_{g,n})\ne\mathcal{U}(\Sigma_{g,n}).$$

The paper notes that this would follow from the non-finite generation conjecture, since finite generation of $\mathcal{A}(\Sigma_{g,n})$ would be forced by equality with $\mathcal{U}(\Sigma_{g,n})$. The genus-one cases are proved in the paper.