Scale and Möbius covariance in two-dimensional Haag–Kastler net

Given a two-dimensional Haag-Kastler net which is Poincare-dilation covariant with additional properties, we prove that it can be extended to a Mobius covariant net. Additional properties are either a certain condition on modular covariance, or a variant of strong additivity. The proof relies neither on the existence of stress-energy tensor nor any assumption on scaling dimensions. We exhibit some examples of Poincare-dilation covariant net which cannot be extended to a Mobius covariant net, and discuss the obstructions.