Injectivity conjecture for the 3d quantum trace map

Let $Y$ be the ideally triangulated $3$-manifold considered in the paper, with triangulation $\mathcal{T}$, and let $\operatorname{Tr}_{\mathcal{T}}$ denote its 3d quantum trace map. 3d quantum trace injectivity conjecture. The map

$$\operatorname{Tr}_{\mathcal{T}}$$

is injective. The source presents this as a consequence of injectivity of $\overline{\sigma}$ together with the cited proposition; no resolution is given.