Completeness conjecture for Liouville conformal blocks

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

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

Liouville conformal blocks are the delta-function normalizable conformal blocks whose internal conformal weights satisfy

$$\Delta\geq\frac{c-1}{24}.$$

Liouville completeness conjecture. Liouville conformal blocks form a complete basis with respect to the inner product above. This is the second mathematical gap in the rigorous definition of the inner product; the paper treats its validity as folklore in physics and assumes it thereafter.