Geometric model conjecture for tropical ${\rm SL}_3$ laminations

Let $\mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^{T})$ be the set of tropical-integer points of the ${\rm SL}_3$ cluster $\mathscr{A}$-space of a generalized marked surface $\frak{S}$. Geometric model conjecture. There is a natural geometric model for the set

$$\mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^{T}).$$

Such a model is an implicit prerequisite for interpreting these tropical points as laminations or related geometric objects; the source gives no further specification or resolution.