LLMs as they currently exist will never yield a true, actually-sentient AI. maybe they will get better in some ways, but it's like asking if a bird will ever fly to the moon. Something else is needed.
That, or they "keep improving every month" til they become evolved enough to build rockets, at which point the entire point of them being birds becomes moot.
It would need to become something literally extraterrestrial that has not evolved in the 3.7b+ years prior.
I wouldn't say it's impossible, but if evolution ever got there that creature would be so far removed from a bird that I don't think we'd recognize it :p