Fock–Goncharov's duality conjecture for ${\rm SL}_3$–${\rm PGL}_3$

Let $\frak{S}$ be a generalized marked surface. The set $\mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^{T})$ consists of tropical-integer points of the $\mathscr{A}$-moduli space, and ${\bf L}(\mathscr{X}_{{\rm PGL}_3,\frak{S}})$ is the ring of rational functions that are Laurent polynomial in every ideal-triangulation cluster chart. Fock–Goncharov's duality conjecture. There exists a canonical map

$$\mathbb{I}:\mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^{T})\to{\bf L}(\mathscr{X}_{{\rm PGL}_3,\frak{S}})$$

with favorable properties, including injectivity, the property that its image is a basis, positive-integer structure constants, and positive integer Laurent coefficients in every ideal-triangulation chart. This is the proposed ${\rm SL}_3$–${\rm PGL}_3$ generalization of the known ${\rm SL}_2$–${\rm PGL}_2$ result; the source does not state a resolution.