Non-finite generation conjecture for cluster algebras of positive-genus surfaces

Let $\Sigma_{g,n}$ be a surface of genus $g$ with $n$ punctures, and let $\mathcal{A}(\Sigma_{g,n})$ denote its cluster algebra. Non-finite generation conjecture. For $g\geq 2$, $\mathcal{A}(\Sigma_{g,2})$ is not finitely generated. This would settle the remaining $n=2$ case in the expected non-finite generation of $\mathcal{A}(\Sigma_{g,n})$ for positive genus; the paper states that no proof is known in these cases.