Doubt it. GenAI requires a shitton of resources, both in storage and for processing. Training a GenAI requires clusters upon clusters of NPUs and/or GPUs, even more than crypto miners and 3D renderers. The full storage requirements are proportional to the amount of training data you give, so expect them at least to be dozens of gigabytes long.

I doubt AI companies do it "for science" (yeah, right) so if they're shit down by a court of law they'll just shut the thing down. They can upload the code somewhere, but without training data their engine is useless.