Holomorphicity conjecture for Liouville fusion coefficients

Let $Q=b+b^{-1}$, and let $\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_{23}$ be complex Liouville momenta. The fusion coefficients are denoted by $F_{\alpha_{21}\alpha_{32}}\bigl[{}^{\alpha_3}_{\alpha_4}{}^{\alpha_2}_{\alpha_1}\bigr]$. Holomorphicity conjecture. The fusion coefficients are holomorphic in the domain

$$\begin{aligned} 0<&\operatorname{Re}(\alpha_2+\alpha_3+\alpha_{23}-Q)<Q\\ 0<&\operatorname{Re}(\alpha_2+\alpha_3-\alpha_{23})<Q\\ 0<&\operatorname{Re}(\alpha_{23}+\alpha_3-\alpha_2)<Q\\ 0<&\operatorname{Re}(\alpha_{23}+\alpha_2-\alpha_3)<Q \end{aligned} \qquad \begin{aligned} 0<&\operatorname{Re}(\alpha_4+\alpha_1+\alpha_{23}-Q)<Q\\ 0<&\operatorname{Re}(\alpha_4+\alpha_1-\alpha_{23})<Q\\ 0<&\operatorname{Re}(\alpha_{23}+\alpha_4-\alpha_1)<Q\\ 0<&\operatorname{Re}(\alpha_{23}+\alpha_1-\alpha_4)<Q. \end{aligned}$$

This analyticity property is motivated by the expected analyticity of conformal blocks and by the consistency equations governing fusion coefficients. The source gives no resolution of the conjecture.