Using work equipment for a personal project only matters because you signed a contract giving all of your IP to your employer for anything you did with (or sometimes without) your employer's equipment.

Anthropic's user agreement does not have a similar agreement.

My point was that they could make a compelling case though, not that they would win.

I don't know of ant precedent where the code was literally generated on someone else's system. Its an open question whether that implies any legal right to the work and I could pretty easily see a court accepting the case.