It is difficult enough to prove intelligence in humans, let alone souls.
With souls defined as a collection of paradigms, models, values and preferences making up a human personality. (Souls are often defined as something more ephemeral, but then how do you explain belief in ghosts retaining their personalities and knowledge if not by being 'lost souls'?)
Even even if you somehow recorded the human superego, I predict (because my theory predicts it) that if 'run' outside of it's original body it will be another mind altogether. Disproving a Soul, but proving that the mind is a manifestation of the flesh.
You can bet the definition of 'soul' will be redefined to mean something else entirely, something even more nebulous and definately not testable (at that point). Until science reaches a point where it is possible to test that new definition as well, then the redefinition game will go for another round.
And more scientific energy is wasted.
It is a sorry pattern in which extremely unlikely claims become more and more nebulous and vague as discussion goes on. In the end it always ends up in "there MUST be something else, just outside of science's reach."
I must repeat my vision of the believer's shortcut here:
Yhe universe is a book with pictures. By refusing to learn to read and only looking at the pretty pictures while making up a story, they may get a more colourful and imaginative story, but not the story that is in the book.
Anybody who can read, can reproduce the same story from the book, it is true. The imagined story, by contrast, exists only in the mind of those who heard it told. We know what happens to stories that get retold; they mutate. it is fantasy.
Also at the same time proof for that the world is not a shared illusion: science is true for everyone. More people believe in things unscientific and believe more strongly. More loudly at any rate.