Dunno for sure. I think that they can. The anticorrelation is rather small, after all. So some people with high trait will occasionally have high state curiosity, of course.

I suspect that some people really like to think of themselves as curious such that feeling that way is a part of their identity. They consider themselves to be generally curious about everything. However, there are almost an infinite number of things you could be curious about. It seems unlikely that even a high trait curiosity person is really curious about everything.

If I construct several curiosity experiments with tasks asking about 1) math and logic problems, 2) fashion in late 15th Century Venice, 3) the mating habits of honeybees, and 4) political conflicts in the Han Dynasty, how many self-reported high curiosity people do you expect would rate all of those tasks with high degrees of curiosity? I suspect not all or even many.

Still, I don't think this means that high trait curiosity doesn't exist [edit: s/exists/doesn't exist/]. I just think it's not a good predictor of curiosity in the task or specific performance, among other things.

Do you know if there is a non-self-reported 'trait curiosity' factor? I.e., regardless of how curious they say they are, are some people just curious about more things than others? And if so, does that correlate with intelligence? That would match my intuition.

I do not know. I believe the literature reports at least some correlation between trait curiosity from surveys/ratings/inventories and intelligence (e.g., IQ tests), so your intuition is probably reasonable.