Modular cocycle conjecture for massive double cone algebras

Let $4\mathcal{A}(\mathcal{O})4$ be the algebra of a massive theory associated with a double cone $4\mathcal{O}4$, and let $4\omega_{m=0}4$ denote the conformally invariant vacuum state of the associated massless theory. The massive and massless local algebras may be identified up to unitary equivalence, while their nets inside $4\mathcal{O}4$ remain different. Modular cocycle conjecture. The modular group of the massive double cone algebra with respect to the massive vacuum is cocycle-related to the known geometric modular group of the associated conformally invariant situation; the massive net inside $4\mathcal{O}4$ may be obtained from the massless $4\mathcal{A}(\mathcal{O})4$ by adjoining the action of Poincaré covariances inside $4\mathcal{O}4$. The cocycle accounts for the difference between massive and massless local propagation, so the modular action is fuzzy away from the horizon, while the geometric conformal transformation reappears asymptotically near the boundary. The source notes that the mechanism can be shown for a massive free Fermi field in $1+1$ dimensions, but a general proof remains desirable.