State-Operator Correspondence and symmetry in CFT in general dimension

Question

Let us assume to have a QFT ($\mathcal{L}$) with translational, Lorentz, scale and conformal invariance.
I ask because we can, for example the free scalar free theory, canonically quantize the system and generate states that don't have scale symmetry (e.g. much like plane waves $a_p^\dagger\left|0\right>$ don't have spherical symmetry).
Are there CFTs that when we quantize can have states that don't have conformal symmetry and can't be written as a sum of states with conformal symmetry?
If the answer is yes, we just disregard them in the state-operator correspondence?

Answer 1

I've got my answer in the comments and answers here https://physics.stackexchange.com/questions/747918/why-is-the-limiting-operator-in-the-cft-state-operator-correspondence-well-defin\ These Dilatation states do not span the entire Hilbert space. So it is just their subset that has the state-operator correspondence.