I think that if they would, that would be very surprising and indicative of a lot of wastefulness inside the model architecture. All these tests are simple single prompt experiments, so the LLM's get no chance to reason about their responses. They're just system 1 thinking, the equivalent of putting a gun to someone's head and asking them to solve a large division in 2 seconds. I bet a lot of these experiments would…
But if a human is allowed time and internal reasoning iterations, so should the LLM when determining if it has deep insight. Right now we're simply observing input -> output of LLMs, the equivalent of snap answers from a human. But nothing says it couldn't instead be an input -> extensive internal dialogue, maybe even between multiple expert models for seconds, minutes or hours, that are not at all visible to the prompter -> final insightful answer. Maybe future LLMs will say, "let me get back to you on that".