Fock–Goncharov's algebraic modular functor conjecture for $\mathrm{PGL}_{n+1}$

Let $G=\operatorname{PGL}_{n+1}$, let $S$ be a marked surface, and let $c$ be an essential simple closed curve on $S$. Let $S'$ be obtained by cutting $S$ open along $c$, and let $S^\circ$ be obtained by shrinking the resulting boundary circles $c_+$ and $c_-$ to punctures $p_+$ and $p_-$. Let $\mathbb L_{G,S}$ and $\mathbb L_{G,S'}$ be the corresponding quantum universal Laurent rings. Let $\mathcal I_c\subset\mathbb L_{G,S'}$ be the central ideal generated by the relations $\chi_{p_+}=\chi^*_{p_-}$, and let $W(c)$ be the diagonal subgroup of $W_{p_+}\times W_{p_-}$ preserving this ideal. Assume that neither $p_+$ nor $p_-$ is the only special point on its connected component of $S'$. Algebraic modular functor conjecture. There is a canonical subalgebra $R_G(c)\subset\mathbb L_{G,S}$ and an algebra isomorphism

$$\eta_c:\mathbb L_{G,S}^{R_G(c)}\simeq\left(\mathbb L_{G,S'}/\mathcal I_c\right)^{W(c)},$$

where $\mathbb L_{G,S}^{R_G(c)}$ is the centralizer of $R_G(c)$ in $\mathbb L_{G,S}$. The map is equivariant for the action of the centralizer $\Gamma_{S;c}$ of the Dehn twist along $c$, and restricts to an isomorphism $R_G(c)\simeq R_G(p)$. This is the proposed algebraic modular-functor behavior of the quantum Teichmüller algebras under cutting and gluing surfaces; the source gives the conjecture with references to Fock–Goncharov and Ganev–Schrader, and does not state a resolution.