Does Conformal Quantum Field Theory Describe the Continuum Limits of 2D Spin Models with Continuous Symmetry?

It is generally taken for granted that two-dimensional critical phenomena can be fully classified by the well known two-dimensional (rational) conformal quantum field theories (CQFTs). In particular it is believed that in models with a continuous symmetry characterized by a Lie group $G$ the continuum theory enjoys an enhanced symmetry $G\times G$ due to the decoupling of right and left movers. In this letter we review the conventional arguments leading to this conclusion, point out two gaps and provide a conterexample. Nevertheless we justify in the end the conventional conclusions by additional arguments.