Existence conjecture for timelike Liouville correlation functions

Let $c\geq 25$ and consider the inner product on conformal blocks

$$\langle \mathcal{F}_1\,|\,\mathcal{F}_2\rangle=\int_{\mathcal{T}_{g,n}} Z_{bc}\,\overline{\mathcal{F}}_1\mathcal{F}_2\,Z_{26-c},$$

where $Z_{bc}$ is the ghost partition function and $Z_{26-c}$ is the timelike Liouville correlation function with central charge $26-c$. Timelike Liouville existence conjecture. The timelike Liouville correlation function entering this inner product exists. This is one of the mathematical gaps in the rigorous definition of the inner product; its validity is regarded by physicists as folklore and is assumed in the paper.