The universal quantum Laurentness conjecture for quantum duality elements

Let $\frak{S}$ be a triangulable punctured surface. A congruent ${\rm SL}_3$-lamination is an element $\ell \in \mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^t)$, and $\mathbb{I}^q(\ell)$ is the corresponding quantum duality element in $\mathscr{O}^q_{\rm tri}(\mathscr{X}_{{\rm PGL}_3,\frak{S}})$. The ring $\mathscr{O}^q_{\rm cl}(\mathscr{X}_{{\rm PGL}_3,\frak{S}})$ consists of elements that are quantum $\mathscr{X}$-Laurent for all cluster $\mathscr{X}$-seeds. Universal quantum Laurentness conjecture. For each congruent ${\rm SL}_3$-lamination $\ell \in \mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^t)$,

$$\mathbb{I}^q(\ell) \in \mathscr{O}^q_{\rm cl}(\mathscr{X}_{{\rm PGL}_3,\frak{S}}).$$

Equivalently, $\mathbb{I}^q(\ell)$ is quantum $\mathscr{X}$-Laurent for all cluster $\mathscr{X}$-seeds for $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$, not only those corresponding to ideal triangulations of $\frak{S}$. This conjecture would extend the established Laurentness from triangulation seeds to all cluster seeds and is the prerequisite for expressing the deformation quantization using the full cluster variety.