The modular functor conjecture for higher quantum Teichmuller theory is closely related to a relatively old conjecture in the representation theory of quantum groups. The latter asserts that the category of positive representations of a quantum group is monoidal. This statement seems particularly interesting because positive representations are manifestly bimodules for a quantum group and its modular (aka quantum Langlands) dual. I will describe connection between the two conjectures and outline a proof of the latter one.

This talk is based on a joint work in progress with Gus Schrader.