That idea assumes consciousness as emergent or made out of the energy and matter we perceive. While the content of our experiences obviously is, I'm not so sure about consciousness "itself". I personally feel it's self-evident the observer must be outside and cannot be made of what is being perceived. Like the dreamer is not made of dream "stuff", the gamer is not made out of video game code, the mirage in the mirror is not the mirror itself, etc...

So yes, "immortality" of "that which is conscious" is a given imho, because you cannot possibly be aware of non-existence (see: the thought experiment of quantum immortality). On the other hand, immortality of the individual and all its facets seems impossible given the constant change/entropy of everything.