${A}_2$-bracelets basis conjecture for ${\rm PGL}_3$ character varieties

Let $\mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^{T})$ be the tropical-integer lamination set, and let $\mathbb{I}^{\rm br}$ be the map obtained by applying the paper's construction to the ${A}_2$-bracelets map. ${A}_2$-bracelets basis conjecture. There is a map

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

which is injective and whose image forms a basis of $\mathscr{O}(\mathscr{X}_{{\rm PGL}_3,\frak{S}})$. The source calls this a conjectural basis and contrasts it with the paper's $A_2$-bangles basis; no resolution is given.