Scale and M\"obius covariance in two-dimensional Haag-Kastler net

Given a two-dimensional Haag-Kastler net which is Poincar'e-dilation covariant with additional properties, we prove that it can be extended to a M"obius 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 Poincar'e-dilation covariant net which cannot be extended to a M"obius covariant net, and discuss the obstructions.