All the proof needed would be to look at a piece of paper behind your head and tell the investigators what's written on it, which should NOT be hard at all.
I know there's been something similar for
OOBEs. Asked (surviving[1]) post-operative patients of major trauma if they had an OOBE and, if so, what they saw. With some pattern on the floor of the operating theatre, or somesuch thing that they would have been able to see from their OOB POV but not, even subconsciously, from their more physical positioning.
I can't recall whether it turned out as a Yay or a Nay, though, on that experiment. Which I believe was singular (at least at the time) and I've a worrying feeling was not entirely cleared with the Ethics Committee.
Fever-induced sensations can be interesting. I was once (when quite young) suffering a bit of restless sleep from whatever it was that I'd got at that time. I'd obviously (in hindsight) been flailing my limbs a bit, sending my bedclothes into paroxysms of motion very much like stormy seas. My consciousness of the situation had me observing this with total disassociation from the fact that I was causing this. I shouted at the bedclothes to "Stop it!"... Only for a few moments later to do a mental face-palm... There was no use telling the bedclothes to stop, as I realised that
they couldn't understand English!(Cue double-mental-face-palm, when I later remembered these events.)
As to the whole... well,
everything... of ESP/etc, my Dad's commented at how many times he's thought about someone/something and then that person/someone associated with the thing phones up. And I know the feeling. I also know that both he and me doubtless think about countless random things, at all times, and the times when a person does
not call after being thought about are never remembered. Not to mention those times when some common cue (something having been on TV, perhaps) has perhaps cued the respective thought/contact scenaio. (As I was contemplating this paragraph, I received an SMS from my Dad. A rare-ish occasion, and without a common-cue. But I don't give it any more relevance than mere happenstance/synchronicity. Maybe it
is significant, but I really can't see how it can be quantified.)
[1] Asking those that
weren't involves a totally different metaphysical proof-of-concept, of course...