Laurent coefficient positivity conjecture for the ${\rm SL}_3$–${\rm PGL}_3$ duality map

Let $\mathbb{I}$ be the classical duality map and let $\mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^{T})$ index its basic functions. Laurent coefficient positivity conjecture. For each $\ell\in\mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^{T})$, the basic regular function $\mathbb{I}(\ell)$ can be written in any cluster $\mathscr{X}$-chart as a Laurent polynomial in the generators with non-negative integer coefficients. This is a positivity strengthening of the duality-basis construction; the source gives no resolution.