The injectivity conjecture for the splitting maps

Let $\sigma$ and $\overline{\sigma}$ be the splitting maps associated with the 3d quantum trace construction. Splitting-map injectivity conjecture. The maps $\sigma$ and $\overline{\sigma}$ are injective. Injectivity of $\overline{\sigma}$ would imply injectivity of the 3d quantum trace map, as stated immediately afterward; the source supplies no resolution.