There is nothing absurd about that liver remaining a proper part of me while gaining a conscious mind. (Certainly this is true if materialism is true—cf. this post.) Then I would have two conscious minds as proper parts of me—the liver's mind and the one with which I now think. Plausibly, however, I would not be aware of what the liver is thinking. I can kind of imagine my liver being a homunculus of whose thoughts I am quite unaware. But, it seems, the liver's mind would be a part of me, and hence would be a mind of mine. And, surely, I should be conscious of what a mind of mine thinks.
So we have problem: if my liver gained a mind, it seems I would both be and not be conscious of what the liver was thinking. Now there is a way of embracing this paradox. I might distinguish as follows. In the hypothetical situation, I would have two minds, A and B. I would be aware of what both are thinking. But I would not be aware with A of what I am thinking with B, and I would not be aware with B of what I am thinking with A.
What is kind of fun is that the above considerations yield an argument for the disjunction of two controversial views, both of which I hold. Suppose you think it is absurd that I should gain a second mind and be unaware with this mind of what that mind is thinking. Then, I think, you need to stop my thought experiment from going through. I think your best bet for stopping my thought experiment from going through is to deny that my liver and my mind are parts of me. Maybe the best way to do this is to insist that I do not have proper parts. (Of course I also have a liver. But it does not follow that livers exist, just as it does not follow from my having had a fright that a fright existed.) If so, then the rejection of where my thought experiment leads to gives a plausible argument for the controversial thesis that I don't have proper parts.
But suppose one embraces the conclusion. Then, one accepts something else controversial that I hold, namely that it is possible for one person to have two minds, and to be unaware with one of what he is thinking with the other. The case I am interested in is that of Christ. He has a human mind and a divine mind. And with his human mind he, probably, cannot be aware of everything that he is divinely thinking. Of course, this may force a denial of the transcendental unity of apperception.