From my experience, their hallucination when playing poker mostly comes from a wrong reading of their hand strength in the current state. E.g., thinking they have the nuts when they are actually on a nut draw. They would reason a lot better if you explicitly give out their hand strength in the prompt.
I noticed the same and think that you're absolutely right. I've thought about adding their current hand / draw, but it was too close to the event to test it properly.
