Positive representations are certain bimodules for a quantum group and its modular dual. In 2001, Ponsot and Teschner constructed these representations for U_q(\mathfrak{sl}_2) and proved that they form a continuous braided monoidal category, where the word "continuous" means that a tensor product of two representations decomposes into a direct integral rather than a direct sum. Ten years later, their construction was generalized to all other types by Frenkel and Ip. Although the corresponding categories were braided more or less by construction, it remained a conjecture that they are monoidal. Following a joint work with Gus Schrader, I will discuss the proof of this conjecture for U_q(\mathfrak{sl}_n). The proof is based on our previous result where the quantum group is realized as a quantum cluster \mathcal X-variety. If time permits, I will outline a relation between this story and the modular functor conjecture in higher Teichmüller theory along with several other applications.