Quantum upper cluster algebra generation conjecture for projected stated ${\rm SL}_n$-skein algebras

Let $\mathfrak{S}$ be a triangulable pb surface, let $\mathcal V$ be the vertex set of the associated quantum seed, and let $\mathcal V_{\mathrm{mut}}$ be its mutable vertices. Write $R\langle A_v^{\pm1}\mid v\in\mathcal V\setminus\mathcal V_{\mathrm{mut}}\rangle$ for the algebra generated over $R$ by inverses of the frozen variables. Under the same assumption as the theorem establishing the skein-algebra inclusion, the exchangeable cluster variables appearing in the cited equations should generate

$$\mathscr U_\omega(\mathfrak{S})$$

as an $R\langle A_v^{\pm1}\mid v\in\mathcal V\setminus\mathcal V_{\mathrm{mut}}\rangle$-algebra. This is the generation formulation equivalent to the conjectured equality between the projected skein algebra, quantum cluster algebra, and quantum upper cluster algebra; it remains open in the source.