I doubt that is so. In practice they might be trained on the real thing, but models generalise pretty well. It is going to be technically possible to train a model on other material (children, nudity and non-CSAM abuse scenes or maybe not even that) and have it generate CSAM.
But even if it was true, that would only make training the model illegal and ethically dubious. We use a tonne of technologies where the creator was legally and morally dubious. It's never been an ongoing issue before. So once the model is created there isn't a good reason to encumber it by how it was created.