The precedent in this case already exists in Midler v. Ford Motor Co., in which when Academy Award nominated actress and singer Bette Midler sued Ford after Ford hired musical impersonators to sing famous songs for their commercials.

The court ultimately ruled in favor of Midler, because it was found that Ford gave clear instructions to the impersonating actress to sound as much like Midler as possible, and the ruling was voices, although not copyrightable, still constitutes their distinct identity and is protected against unauthorized use without permission. (Outside of satire, of course, since I doubt someone like Trump would be above suing people for making fun of him.)

I think Scarlett Johansson has a case here, but it really hinges on whether or not OpenAI actively gave the instruction specifically to impersonate Scarlett's voice in "Her", or if they used her voice inside the training data at all, since there is a difference in the "Sky" voice and the voice of Scarlett Johansson.

But then again, what do I know, I'm just here to shitpost and promote "Barbie".