... As I said, I don't think such evidence has actually been presented. But it is not difficult to think of examples.
Let's say, for instance, that the evidence Carl Jung cited for a "racial memory" actually held up to scrutiny. We could isolate individuals and demonstrate that they do substantially better if given last Sunday's NYT crossword than when given next Sundays. We can generate a random string of words or colors and teach them to Population A and members of Population B will be able to spontaneously recite them. There could very well be evidence for a transfer of information between individuals that cannot be accounted for physically.
As another example, let's say that we were able to map the brain molecule-for-molecule and determined that the exact same physical state of the brain existed when people were thinking/doing widely different things. Or that we can map a person's brain exactly, wait a week, restore the brain to its mapped state, and the person still remembers what happened in the intervening week. Or any other evidence that the brain is actually a "radio" receiving information and/or control signals from something that is not directly detectable other than through its influence on the brain.
A third example would be if claims about ghosts were substantiated - if there were phenomena that could be thoroughly investigated and were best explained by the intervention of an intelligence that does not appear to have a physical body.
We can come up with cases like this, where it would be appropriate to refer to the phenomenon as "immaterial." As information or mind that does not have a physical medium underlying it. And while no such phenomena have held up to physical scrutiny, it would be bad science to claim that they could not in principle do so.
I find myself at odds with myself. That is, I have argued your point before; i.e., that it is hard to rule some things out
in principle. Unfortunately, those examples, all really relating to a soul-like self that is not dependent on matter and energy or hosts itself by other means while surfing the ether, can be ruled out. More on that later.
What can't be ruled out entirely is some possible working of some purported immaterial realm that, by definition is not observable. In a word, this is solely a matter of faith, or pure conjecture. As it is nothing that ever impinges on this world observably, who cares? No way to establish the veridical nature of any statement. But also hard to establish that no particle of any kind has any mediating effect on some non-observable medium, since there is no peeking into woo-land.
Back to the examples. They seem to share a view that consciousness is a matter of fine granularity, as if an extra 'mind' particle permeates the brain. This is not so; google for medical definitions of consciousness and brain states, and you will see they are quite macro. That is, no consciousness obtains in the absence of certain macro features, end of story. QM et al is not explanatory.
Returning to the very old debate on ruling out fully immaterial things - or purest woo, it was said in an old thread, quoting Sean Carroll, that QFT ruled everything out; all forces were accounted for, so no 'woo carrying forces' could exist. It occurs to me that that argument is incomplete, as it deals only with baryonic matter. Iirc, there are some theories out there today positing mediating particles in dark matter, called dark photons. Not that I'm peddling that as any sort of woo carrier by any means, or making any claim whatsoever (heaven forbid), but it does go to show that it is hard to close the door on new discovery just yet.
And back to the brain and mind: no matter how many new particles or fields, since consciousness is not at that scale, no immortal soul will ever come of new discoveries in physics.