Higher-genus generalization of the Toeplitz operator result in TQFT

Let $\mathcal M(\Sigma,t)$ be the moduli space associated with a surface $\Sigma$ and parameter $t$. In the higher-genus setting, model $\mathcal M(\Sigma,t)$ by a toric manifold, with its singularities replacing the two poles of the sphere and its Laplacian replacing the Laplacian on the sphere. Higher-genus generalization conjecture. Result 2 is true in the general higher-genus case under these replacements. This conjecture proposes extending the Toeplitz-operator and geometric-quantization analysis from the sphere case to higher-genus moduli spaces. The paper gives partial results for genus $2$, while the general higher-genus case remains open.