The quantum-double realization conjecture for completely rational conformal nets

Let $\mathcal{F}$ be a unitary fusion category, and let $Z(\mathcal{F})$ denote its Drinfeld center. For a finite-depth subfactor $N\subset M$, let $D(N\subset M)$ denote its quantum double. Quantum-double realization conjecture. For every unitary fusion category $\mathcal{F}$, there is a completely rational conformal net $\mathcal{A}$ such that

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

Equivalently, for every finite-depth subfactor $N\subset M$, there is a completely rational conformal net $\mathcal{A}$ such that

$$\operatorname{Rep}(\mathcal{A})\cong D(N\subset M).$$

This is a weaker consequence of the UMTC realization conjecture and concerns realizing Drinfeld centers, or quantum doubles of subfactors, in chiral conformal field theory. A general construction from subfactors or fusion categories to conformal field theories was not established in the source, although the paper discusses recent approaches and proves realizations for subfactors of index less than $4$.