They have to prove it but if they case gets far enough they will have the right to ask for discovery and they can see for themselves what was included. Thats why it might just settle quietly to avoid discovery.

Right but you can sue for what happened on the training server. I'm guessing the training server still exists. I doubt they wiped it completely before the next round of training. If the training server infringes copyright then you still lose the suit. Maybe. Remember that copyright law is not written with the internet in mind. If you have a "copy" and it's not authorized that might just be enough for a backwards court to find infringement.

I think of it in extremes. Imagine you had a video producing model of the future. Could you then load up every MLB game recorded and train the model to make novel baseball games based on that or would the MLB be pissed you had a server full of every MLB game ever recorded?

But the server used to calculate the model would have a copy of it. If training an AI model is not fair use then the mere act of loading a book you don't have a license for into the server would be copyright infringement. Like text book. It's a unauthorized digital copy. It's all very untested legal grounds and seems like lots of people want to be the first to test it. Not everyone has a great case but if the courts interpret things a certain way there's gonna be lots of payouts so maybe best to get in line early?