Cyclic subcategories beyond the known example classes

Identify examples of cyclic subcategories of module categories over a vertex operator algebra that do not belong to the finite, C2-cofinite, finite rigid, or non-finite rigid example classes described in the paper.

Background

The paper constructs modular functors with singularities from a cyclic subcategory of modules over a vertex operator algebra. It develops several sources of such cyclic subcategories, including finite categories, C2-cofinite module categories, finite rigid tensor categories, and certain non-finite rigid settings.

The authors leave unresolved whether further examples exist outside these classes. Finding such examples would broaden the range of vertex operator algebras to which the construction applies and clarify how common cyclic structures are in representation theory.