I have a vague intuition – most likely incorrect – that all (internal) flavor symmetries in a CFT are always spontaneously broken. (Perhaps under some mild assumptions, such as unitarity and that the conformal symmetry itself is unbroken).
Is this actually provable? Or are there any known counterexamples (i.e., a CFT with a faithful global symmetry that is unbroken)? Are there counterexamples for both continuous and discrete symmetries?
I mostly care about $d>2$ since Coleman–Mermin–Wagner won't allow continuous symmetries to break in 2d. But of course if there is anything known about 2d specifically I would also like to hear it.