The admissible-genus realization conjecture for unitary modular categories

Let $\mathcal{C}$ be a unitary modular category, and let $(\mathcal{C},c)$ denote an admissible genus consisting of $\mathcal{C}$ together with a compatible central charge $c$. Admissible-genus realization conjecture. There is a central charge $c$ such that the admissible genus $(\mathcal{C},c)$ is realizable. The source gives no resolution and places the claim in the problem of constructing chiral CFTs from their representation modular categories.