Quantum Laurent coefficient positivity conjecture

Let $\Delta$ be an ideal triangulation of $\frak{S}$ and let $\mathbb{I}^q_\Delta$ be the proposed quantum duality map. Quantum Laurent coefficient positivity conjecture. For every $\ell\in\mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^{T})$, the element $\mathbb{I}^q_\Delta(\ell)$ is a non-commutative Laurent polynomial in $\{\widehat{X}^{\pm1}_v\mid v\in\mathcal{V}(Q_\Delta)\}$ with coefficients in

$$\mathbb{Z}_{\geq0}[q^{\pm1/18}].$$

This is the quantum analogue of Laurent coefficient positivity, with emphasis on non-negative coefficients; the source gives no resolution.