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×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 counterexample. Nevertheless we justify in the end the conventional conclusions by additional arguments.