In rational conformal field theory, a main conjecture is that the (unitary) modular tensor category associated to a (unitary) vertex operator algebras (VOA) $V$ is equivalent to the one associated to the corresponding conformal net $\mathcal {A}_{V}$ (Kawahigashi in J. Phys. A, Math. Theor. 48(30):303001, 2015; Kawahigashi in Conformal field theory, vertex operator algebras and operator algebras. Proceedings of ICM-2018, 2018, arXiv:1711.11349). In (Gui in Commun. Math. Phys. 383:763–839, 2021), we gave a systematic treatment of this conjecture by introducing the notion of categorical extensions of conformal nets. One major difficulty encountered in that article is to prove the strong braiding of smeared intertwining operators of $V$ . In this article, we develop a theory of unbounded operators in categorical extensions of conformal nets, and show that strong braiding follows from some other conditions which are much easier to verify. As an application, we prove the equivalence of unitary modular tensor categories of $V$ and $\mathcal {A}_{V}$ for the following examples: all WZW models, all even lattice VOAs, all parafermion VOAs with positive integer levels, type $ADE$ discrete series $W$ -algebras, their tensor products, and their regular coset VOAs. In the case of WZW models, our work together with (Finkelberg in Geom. Funct. Anal. 6(2): 249–267, 1996) solve the longstanding conjecture that given a simple Lie algebra with positive integer level, the unitary modular tensor categories associated to the corresponding affine Lie algebra, quantum group at certain roots of unity, and loop group conformal net are equivalent.
