Yeah, but I wouldn't count "Which prompt makes you more truthful and logical" amongst those.

The questions it will always confabulate are those that are unknowable from the training data. For example even if I give the model a sense of "identity" by telling it in the system prompt "You are GPT6, a model by OpenAI" the training data will predate any public knowledge of GPT6 and thus not include any information about the number of parameters of this model.

On the other hand "How do I make you more truthful" can reasonably be assumed to be equivalent to "How do I make similar LLMs truthful", and there is lots of discussion and experience on that available in forum discussions, blog posts and scientific articles, all available in the training data. That doesn't guarantee good responses and the responses won't be specific to this exact model, but the LLM has a fair chance to one-shot something that's better than my one-shot.