Curve and upper cluster algebra comparison conjecture for punctured surfaces

Let $\Sigma_{g,n}$ be a surface of genus $g$ with $n$ punctures, let $\mathcal{C}(\Sigma_{g,n})$ be its curve algebra, let $\mathcal{C}(\Sigma_{g,n})'$ be the subalgebra generated by the relevant geometric elements, and let $\mathcal{U}(\Sigma_{g,n})$ be the upper cluster algebra. Curve–upper algebra comparison conjecture.

$$\mathcal{C}(\Sigma_{g,1})=\mathcal{U}(\Sigma_{g,1}).$$

If $n\geq 2$,

$$\mathcal{C}(\Sigma_{g,n})'=\mathcal{U}(\Sigma_{g,n}).$$

In particular, if $n\geq 4$,

$$\mathcal{A}(\Sigma_{0,n})=\mathcal{U}(\Sigma_{0,n}).$$

These comparisons are presented as a conjectural summary; the source remarks that the relevant equality was not rigorously proved for the general case.