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

Let $\mathscr{A}_{\rm L}(\frak{S};\mathbb{Z})$ be the set of ${\rm SL}_3$ laminations and let $\mathbb{I}^{\rm br}_{{\rm SL}_3}$ be the map obtained by replacing loop components with bracelet-type trace functions. ${A}_2$-bracelets basis conjecture. The map $\mathbb{I}^{\rm br}_{{\rm SL}_3}$ is injective, and its image forms a basis of

$$\mathscr{O}(\mathscr{X}_{{\rm SL}_3,\frak{S}}).$$

The source says this should follow from an analogous skein-algebra statement via an upper-triangular base change, but records it as a conjecture.