Liouville fusion-transform conjecture for conformal blocks

Let $\mathbb S$ be the contour of intermediate Liouville momenta, and let ${{\mathcal F}}^s_{\alpha_{21}}$ and ${{\mathcal F}}^t_{\alpha_{32}}$ denote the s- and t-channel conformal blocks for four external momenta $\alpha_1,\alpha_2,\alpha_3,\alpha_4$. The fusion coefficients are denoted by $F_{\alpha_{21}\alpha_{32}}\bigl[{}^{\alpha_3}_{\alpha_4}{}^{\alpha_2}_{\alpha_1}\bigr]$. Fusion-transform conjecture. There exist invertible fusion transformations between the s- and t-channel conformal blocks:

$${{\mathcal F}}^s_{\alpha_{21}}\bigl[{}^{\alpha_3}_{\alpha_4}{}^{\alpha_2}_{\alpha_1}\bigr]({\mathfrak z})=\int\limits_{\mathbb S}d\alpha_{32}\;F_{\alpha_{21}\alpha_{32}}\bigl[{}^{\alpha_3}_{\alpha_4}{}^{\alpha_2}_{\alpha_1}\bigr]{{\mathcal F}}^t_{\alpha_{32}}\bigl[{}^{\alpha_3}_{\alpha_4}{}^{\alpha_2}_{\alpha_1}\bigr]({\mathfrak z}).$$

Such transformations are the basic ingredient needed to establish locality, or crossing symmetry, of Liouville correlation functions; the source presents their existence as a fundamental assumption rather than proving it.