Joffrey wasn't cold before Dontos came up and said, let's go. He takes her to the river and Littlefinger is conveniently standing there with a boat. It was like he was waiting on her to arrive, so I can see how non-readers came to that conclusion.

Also, I think the Tyrells laced the cup when they gifted it to Joffrey making the necklace a non-issue in his death.