$\mathcal{A}=\mathcal{U}$ for cluster algebras from moduli spaces of $G$-local systems

For a finite-dimensional simple Lie algebra $\mathfrak{g}$ admitting a non-trivial minuscule representation and a connected marked surface $\mathrm{\Sigma }$ with at least two marked points and no punctures, we prove that the cluster algebra ${\mathcal{A}}_{\mathfrak{g},\mathrm{\Sigma }}$ associated with the pair $(\mathfrak{g},\mathrm{\Sigma })$ coincides with the upper cluster algebra ${\mathcal{U}}_{\mathfrak{g},\mathrm{\Sigma }}$. The proof is based on the fact that the function ring $\mathcal{O}({\mathcal{A}}_{G,\mathrm{\Sigma }}^{\times })$ of the moduli space of decorated twisted $G$-local systems on $\mathrm{\Sigma }$ is generated by matrix coefficients of Wilson lines introduced in [IO20]. As an application, we prove that the Muller-type skein algebras $\mathscr{S}_{\mathfrak{g}, \Sigma}[\partial^{-1}]$[Muller,IY23,IY22]for$\mathfrak{g}=\mathfrak{sl}_2, \mathfrak{sl}_3,$or$\mathfrak{sp}4$areisomorphictotheclusteralgebras$\mathscr{A}{\mathfrak{g}, \Sigma}$.