Localized stated skein algebra and quantum cluster algebras conjecture

Let $\mathfrak{S}$ be a marked surface satisfying the assumptions of Theorem~, and let ${\mathscr S}_\omega^{\rm fr}(\mathfrak{S})$ denote the localization of the stated $\operatorname{SL}_n$-skein algebra at the multiplicative subset generated by its frozen variables. Let $\mathscr{A}_{\omega}^{\rm fr}(\mathfrak{S})$ and $\mathscr{U}_{\omega}^{\rm fr}(\mathfrak{S})$ be the corresponding localized quantum cluster and upper cluster algebras.

Localized skein–cluster conjecture. Under these assumptions,

$${\mathscr S}_\omega^{\rm fr}(\mathfrak{S})= \mathscr{A}_{\omega}^{\rm fr}(\mathfrak{S}) =\mathscr{U}_{\omega}^{\rm fr}(\mathfrak{S}).$$

This is the conjectured full equivalence between the localized stated $\operatorname{SL}_n$-skein algebra and the quantum cluster and upper cluster algebras; the supplied text does not give a resolution status beyond presenting it as a conjecture.