Injectivity conjecture for the Panitch–Park quantum gluing module map

Let $\mathcal{T}$ be a triangulation, let $R$ be the coefficient ring, and let

$$\mu\colon\hat{\mathcal{G}}_{\mathcal{T}}\longrightarrow\hat{\mathcal{G}}_{\mathcal{T}}^{[\text{PP}]}$$

be the surjective $R$-module homomorphism from the quantum gluing module to the Panitch–Park quantum gluing module. Injectivity conjecture. The map $\mu$ is also injective, and hence is an isomorphism of $R$-modules. The conjecture would identify the two constructions of the quantum gluing module; the source establishes surjectivity but leaves injectivity as the conjectural part.