Laurent positivity conjecture for the ${A}_2$-bracelets basis

Let $\mathbb{I}^{\rm br}$ be the conjectural ${A}_2$-bracelets basis map for $\mathscr{O}(\mathscr{X}_{{\rm PGL}_3,\frak{S}})$, and let $\mathbb{I}$ denote the paper's corresponding basis map. Laurent positivity conjecture. The map $\mathbb{I}^{\rm br}$, or equivalently $\mathbb{I}$, satisfies the Laurent coefficient positivity property: its basic functions have Laurent expansions with non-negative integer coefficients in every cluster chart. This is the bracelets-basis version of the preceding Laurent coefficient positivity conjecture; the source gives no resolution.