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

Let $\mathbb{I}^{\rm br}$ and $\mathbb{I}$ denote the conjectural bracelets and bangles basis maps, respectively, and let their structure constants be defined by multiplication in the corresponding basis. Structure-constant positivity conjecture. The map $\mathbb{I}^{\rm br}$, or $\mathbb{I}$, satisfies structure-constant positivity: the structure constants of the basis are non-negative. The source poses the corresponding positivity question for the bangles basis and gives no resolution for either map.