Quantum Laurent integrality conjecture for the ${\rm SL}_3$ duality map

Let $\Delta$ be an ideal triangulation of a triangulable generalized marked surface $\frak{S}$, let $\mathbb{I}^q_\Delta$ be the proposed quantum duality map, and let $\mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^{T})$ be its tropical-integer indexing set. Quantum Laurent integrality conjecture. For every $\ell\in\mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^{T})$, $\mathbb{I}^q_\Delta(\ell)$ is a non-commutative Laurent polynomial in $\{\widehat{X}^{\pm1}_v\mid v\in\mathcal{V}(Q_\Delta)\}$ with coefficients in $\mathbb{Z}[q^{\pm1}]$. The source explains that this follows from the preceding exponent congruence conjecture, while its status is otherwise unresolved there.