A controlled study showing that intuition is useless.
First, you'd have to define intuition. As I've said, no neuroscientist denies that humans have a remarkable ability to recognize patterns (especially faces) that goes beyond their ability to explain how they recognize them. (Note: I do not mean to say that neuroscience can't explain how this works, rather that we can recognize patterns when we can't verbalize how we're doing it.) Similarly, many people can recognize musical notes without measuring the frequency of the tone.
That definition of "intuition" is entirely testable, but I don't think there is anyone who doubts the utility of that ability. Now if you claim intuition is something like receiving ideas that come from sources external to the mind, then you have to clarify what you mean.
However, the claim of "synchronicity" is that there is something other than normal probability resulting in low probability outcomes.
How do you propose to investigate this? (When you suggested we need to investigate this, you were talking about synchronicity, not intuition.)
And answering "a controlled study testing the existence of synchronicity" is not really an answer. How exactly do you set up a controlled study?
I believe that any outcome you get will be the same outcome you'd get if there is no such thing as synchronicity. So far, you haven't been able to answer the question I've asked repeatedly--how do you distinguish between a low probability event that is merely a coincidence and one that is synchronicity.
At first you tried to claim that it was just whether or not a person claims the event has meaning or significance, but you took that back when you realize that it makes no sense. (It's just circular reasoning.) You still claim that it's something to do with probability, but you've yet to distinguish a low probability outcome that is mere coincidence from one that is synchronicity.
I think your inability to do this means it is utterly impossible for you to set up any kind of controlled study that will make the case for or against synchronicity.
About the best question you've asked is "What are the odds of that happening?" but you refuse to listen to the answer (that is, the odds of that outcome are the same as the odds of other low probability outcomes that you don't think are cases of synchronicity).