Better, but you need to be tighter than that!
Regardless of intent, it's still possible for Patty and Jack to exist in one of the rooms we looked through and we missed their presence.
Regardless of the actual source of the crimes, it's still possible for Patty and Jack to perceive us as a possible culprit and thus a threat. Alternately, they could have something else they wish to hide and did not wish to be seen, viewing any persons as possible threats to their secrets or safety.
Regardless of whether poison was actually ingested, Zack was poisoned. Perhaps he inhaled it or had it injected.
If not poison, it could be one of numerous documented medical diseases. Under strange circumstances, it could also be a new disease that we don't know about.
If not a disease, he could be faking his illness for some reason. I don't need to explain the reason!