The UMTC realization conjecture for completely rational conformal nets

A unitary modular tensor category (UMTC) is a unitary braided fusion category with non-degenerate braiding. For a completely rational conformal net $\mathcal{A}$, let $\operatorname{Rep}(\mathcal{A})$ denote its category of Doplicher--Haag--Roberts representations. UMTC realization conjecture. For every unitary modular tensor category $\mathcal{C}$, there is a completely rational conformal net $\mathcal{A}$ such that

$$\operatorname{Rep}(\mathcal{A})\cong\mathcal{C}.$$

The conjecture asks whether every UMTC can arise from chiral conformal field theory. The analogous realization problem in higher-dimensional algebraic quantum field theory is known under natural assumptions, but the conformal-net realization asserted here is not established in general.