Finiteness conjecture for conformal nets with fixed modular tensor category

Fix a modular tensor category $\mathcal C$ and a central charge $c$. Consider local conformal nets whose representation category is $\mathcal C$ and whose central charge is $c$. Finiteness conjecture. There are only finitely many such local conformal nets. This is an operator-algebraic finiteness problem for conformal nets with prescribed categorical and central-charge data; the source gives no resolution.