Roger–Yang injectivity conjecture for the curve algebra representation

Let $\Sigma_{g,n}$ be a punctured surface, let $\mathcal{C}(\Sigma_{g,n})_{\mathbb{C}}$ be its complexified curve algebra, and let $\mathcal{T}^{d}(\Sigma_{g,n})$ be its decorated Teichmüller space. Roger and Yang constructed a Poisson algebra homomorphism

$$\Phi:\mathcal{C}(\Sigma_{g,n})_{\mathbb{C}}\to C^{\infty}(\mathcal{T}^{d}(\Sigma_{g,n})),$$

which sends loops, arcs, and vertices to their corresponding $\lambda$-length functions. Roger–Yang injectivity conjecture. The Poisson algebra homomorphism $\Phi$ is injective. This asserts that the curve-algebra relations capture all relations among the corresponding $\lambda$-length functions; the source presents it as a conjecture of Roger and Yang.