Existence of non-local real one-dimensional modular functors

Determine whether non-local real one-dimensional modular functors exist whose deformation-deformation cocycles represent the nontrivial cohomology classes associated with the real parts of the relevant cocycles, and thereby determine whether locality is necessary for the universality theorem.

Background

The paper classifies local Frölicher-smooth real one-dimensional modular functors with a given central charge. In the relative cohomology governing deformation-deformation cocycles, locality eliminates certain additional cohomology classes, leaving the central charge as the sole coefficient. The authors note that without locality, these excluded classes might occur, but they do not determine whether corresponding non-local modular functors actually exist.

Resolving this question would clarify whether the locality hypothesis is merely technical or is genuinely necessary for the stated universality result.