The \mathrm{SL}_n-quantum trace map

Fix a $n^2$-root $\omega=q^{1/n^2}\in\mathbb{C}-\{0\}$. For each ideal triangulation $\lambda$ of the punctured surface $\mathfrak{S}$, let $\mathscr{S}^q_n(\mathfrak{S})$ be the skein algebra and $\mathscr{T}_n^\omega(\lambda)$ the corresponding quantum torus. The $\mathrm{SL}_n$-quantum trace map. There exists an injective algebra homomorphism

$$\mathrm{Tr}^\omega_\lambda:\mathscr{S}^q_n(\mathfrak{S})\longhookrightarrow\mathscr{T}_n^\omega(\lambda)$$

such that, when $q=\omega=1$, every blackboard-framed oriented knot $K$ in $\mathfrak{S}\times(0,1)$ projecting to an immersed closed curve $\gamma$ satisfies

$$\mathrm{Tr}^1_\lambda(K)=\widetilde{\mathrm{Tr}}_\gamma(X_i^{1/n})\in\mathscr{T}_n^1(\lambda).$$

The map is intended to provide a quantum analogue of the classical trace functions, translating skein-algebra elements into the Fock–Goncharov quantum character variety. The supplied text does not state whether this conjectural formulation has been proved, so its database status remains open.