Web parametrization conjecture for positive tropical PGL$_3$ points

Let $\hat{S}$ be a compact oriented surface with marked boundary points and punctures. Let $\mathcal{P}_{{\rm PGL}_3,\hat{S}}^+(\mathbb{Z}^t)$ denote the integer tropical points satisfying the potential condition on every boundary interval and having dominant-coweight monodromy around each puncture, and let $\mathscr{W}_{\hat{S}}^{\mathcal{X}}$ be the set of reduced $\mathcal{X}$-webs on $\hat{S}$. Web parametrization conjecture. There is a natural bijection

$$\mathcal{P}_{{\rm PGL}_3,\hat{S}}^+(\mathbb{Z}^t)\stackrel{\sim}{=}\mathscr{W}_{\hat{S}}^{\mathcal{X}}.$$

This would identify positive integral tropical data for the PGL$_3$ moduli space with reduced $\mathcal{X}$-webs, providing a combinatorial model for the tropical points. The source gives no evidence of a resolution.