Divisibility conjecture for surface localization monoids

Let $\Sigma$ be a marked surface, let $\Delta$ be a triangulation, and let $\mathcal{A}_\Sigma$ be the associated algebra. The natural homomorphism

$$\mathcal{A}_\Sigma\to\operatorname{Frac}(\mathbb{Q}\mathbb{T}_\Delta)$$

comes from the inclusion of $\mathbb{Q}\mathbb{T}_\Delta$ into its fraction skew field. Let $S_\Delta$ denote the submonoid of $\mathbb{Q}\mathbb{T}_\Delta\setminus\{0\}$ generated by the elements inverted in constructing $\mathcal{A}_\Sigma$. Surface localization conjecture. For each $\Sigma$, this homomorphism is injective; equivalently, the submonoid $S_\Delta$ is divisible in the sense of the source's definition. This would establish that the universal localization embeds in the fraction skew field; the source proposes it without stating a resolution.