Uniqueness conjecture for the holomorphic central-charge-24 conformal net

A local conformal net is holomorphic when its only irreducible representation is the vacuum representation. Consider a holomorphic local conformal net with central charge $c=24$, and let $L_0$ be the conformal Hamiltonian. Holomorphic central-charge-24 uniqueness conjecture. If the eigenspace of $L_0$ with eigenvalue $1$ is $0$, then the local conformal net is unique up to isomorphism. This is the operator-algebraic counterpart of the uniqueness conjecture for the Moonshine vertex operator algebra; the source gives no resolution.