The UTMF realization conjecture for Drinfeld centers

Let $V$ be a unitary topological modular functor (UTMF), and let $\mathcal{B}$ be its associated unitary modular category. Let $\mathcal{C}$ be a unitary fusion category. UTMF realization conjecture. $V$ can be realized by a topological phase of matter if and only if

$$\mathcal{B}=\mathcal{Z}(\mathcal{C}).$$

Here $\mathcal{Z}(\mathcal{C})$ is the quantum double, or Drinfeld center, of $\mathcal{C}$. The source notes that the forward construction is expected from the Levin–Wen model, but a fully rigorous proof remains absent.