Allegretti–Kim's structure-coefficient positivity conjecture for quantum elements

Let $S$ be a punctured surface, and let $\mathcal{A}_{{\rm SL}_2,S}(\mathbb{Z}^t)$ denote the set of integral laminations indexing the Allegretti–Kim quantum elements $\widehat{\mathbb{I}}^q(\ell)$. For $\ell,\ell'\in\mathcal{A}_{{\rm SL}_2,S}(\mathbb{Z}^t)$, write their product in the quantum basis as

$$\widehat{\mathbb{I}}^q(\ell)\widehat{\mathbb{I}}^q(\ell')=\sum_{\ell”\in\mathcal{A}_{{\rm SL}_2,S}(\mathbb{Z}^t)}c^q(\ell,\ell';\ell”)\,\widehat{\mathbb{I}}^q(\ell”).$$

Structure-coefficient positivity conjecture. Only finitely many structure coefficients $c^q(\ell,\ell';\ell”)$ are nonzero, and every one belongs to $\mathbb{Z}_{\geq 0}[q,q^{-1}]$; equivalently, each is a Laurent polynomial in $q$ with positive integral coefficients.

The classical structure coefficients are known to be positive integers, and the quantum product is already known to be a finite linear combination with coefficients in $\mathbb{Z}[q,q^{-1}]$. The conjecture asserts that this positivity persists in the quantum setting.