They're not there yet. Once they get there, that's literally the definition of Singularity.
But they are getting closer. Recursive Self-Improvement used to be a phrase people mocked LessWrong crowd for using and worrying about, now it's something both OpenAI and Anthropic already publicly admitted not only to pursue, but to already be benefiting from.
So far, I don't think the models are capable of running away on their own. Of course, it would be playing with fire to not at least consider the risks of such a runaway scenario and build in safeguards against it. But, there is no model that can build a better model on its own, thus far, to the best of my knowledge (which is far more limited than the models, so maybe I should ask them).
It starts with what they already claim to be doing - increasingly relying on existing models in non-trivial work related to training, evaluating and optimizing the next, more capable generation of models. As long as the proportion of work keeps shifting towards agents doing more and more of it, and humans less and less, that's RSI at play.
It may be that it turns out LLMs lack some fundamental level of judgement and it plateaus, but frankly I find this notion absurd; LLMs already show better judgement than most people. The alternative is, at some point LLMs will show the ability to futz their way into improvement of the next generation of models even without humans in the loop - even if much less efficient at first, if generation N+1 is more capable than generation N, it'll either take off or burn out.
At least they’re led by trustworthy and honest people or we’d need to take their claims with some dose of skepticism.