I’ve looked a bit and haven’t seen any experiments at all.
If by control you mean control group, there’s no need for that. If even a single pair of people could pass messages in a way that’s inexplicable except for the DMT realm, having another group bomb the test doesn’t change anything.
I don’t think it’ll ever happen though. I have this belief that something prevents these things getting pinned down by empiricism. Like if you’re caught trying to prove it, the signal gets jammed. Yes, this makes it unfalsifiable. I’ll respect anyone who loses interest for that reason. Unfalsifiable isn’t quite the same as false, though.
Most experiments around actually interacting with entities seem incredibly naive. Like imagine I told you I had a friend who makes the most amazing animal impressions in the world, but he lives in a dangerous neighborhood. One person refuses to go because it’s hazardous, and declares he’s not real. Another walks in, shoves a tape recorder in his face and says “bark like a dog for 10 seconds”. He analyzes the recording and declares it a 0% match. (It was a 100% match for a human saying “piss off” though). But you keep hearing these stories about people who show up, ask nicely, and get to hear the cool animal sounds.