Equivalence of Liouville conformal blocks and quantum Teichmüller quantization

Prove that the quantization of Teichmüller space given by Liouville conformal blocks is equivalent to the Chekhov–Fock–Kashaev quantum Teichmüller quantization, thereby establishing the Verlinde–Teschner correspondence.

Background

Conformal blocks in Liouville theory are holomorphic sections over Teichmüller space and are conjectured to define a quantization equivalent to the one constructed by Chekhov–Fock and Kashaev via hyperbolic geometry.

Verlinde conjectured this equivalence, and Teschner provided supporting arguments; a rigorous proof would unify geometric and CFT quantizations of Teichmüller space.