The tropical parametrization conjecture for the quantum lamination basis

Let $\mathfrak{sl}_3$-laminations with pinnings on a marked surface $\Sigma$ form the integral parameter space $\mathcal{L}_{\mathfrak{sl}_3}^p(\Sigma,\mathbb{Z})$. Let $\mathbb{I}_{\mathcal{X}}^q$ be the quantum trace map, let ${A_i}_{iin I}$ be the quantum cluster associated with a vertex $\omega$ of the exchange graph, and let $x^{(\omega)}=(x_i)_{iin I}$ denote the corresponding negative shear-coordinate system.

Tropical parametrization conjecture. The basis $\mathbb{I}_{\mathcal{X}}^q(\mathcal{L}_{\mathfrak{sl}_3}^p(\Sigma,\mathbb{Z}))$ is parametrized by tropical points. Namely, for every integral $\mathfrak{sl}_3$-lamination $\widehat{L}$, the quantum Laurent expression of $\mathbb{I}_{\mathcal{X}}^q(\widehat{L})$ in the quantum cluster has leading term

$$\prod_{i\in I}A_i^{x_i(\widehat{L})}$$

with respect to the dominance order.

This is the expected connection between the lamination parametrization and the tropical-point framework underlying quantum Fock--Goncharov duality. The supplied text does not state whether the conjecture has been resolved.