Converse rationality question for RCFTs

Ascertain whether the converse of a standard RCFT property holds: if a two‑dimensional conformal field theory has rational central charge and rational conformal dimensions, does it necessarily define a rational conformal field theory (i.e., have only finitely many irreducible representations)?

Background

It is known that in RCFTs both the central charge and conformal dimensions are rational. The converse—whether rationality of c and operator dimensions forces rationality of the theory (finite number of primaries)—is a fundamental structural question.

Resolving this would sharpen the boundary between rational and irrational CFTs and inform classification efforts.