Not how that works. It's like saying show me the study where they prove god doesn't exist.
I think saying "it's a computer program it can't be sentient" is far too reductive as a general answer, but we know how llms work so we can pretty confidently say that they are really just word predictors. The norns from that game creature someone described in this thread seem closer to protection worthy than llms to me, and the average insect is still infinitely more complex and I don't see many people advocating for insect rights.
Language arises from complex thought, not the other way around. Language on its own can evidently emulate similar results, but since there is nothing behind that that would imply any actual feelings, I don't see a reason to give LLMs any rights.