Is science going to bridge the gap between self awareness and our physically material minds?
The basic question boils down to the impossible proposition that science can eventually answer all questions related to cosmic existence.
At some point there is a horizon that science cannot cross, for example science will never prove where the universe came from beyond some, yet undiscovered, event horizon. This also applies to the very origins of life itself, the boundary here is how life started from stochastic chemistry. No matter how far we stretch our knowledge of both they will forever remain empirically unsolved.
New theoretical aspects of consciousness are not adequate and will never be adequate to explain self awareness . Admittedly the findings and theoretical attempts so far verify human beings are indelibly tied to a physicality in our cosmic experience. This only augments our understanding of Biblical concepts but falls short in replacing the body mind and Spirit (Soul) triple alliance notion found in the Bible. Self awareness, according to the Bible, is then the result of an inseparable bond between these three constructs, combined they constitute a self aware "eternal conciseness" not a physical one, explicitly stated this eternal conciseness finds a home in the physical brain.
Will AI ever experience self awareness? I can confidently answer no in terms of Christianity, you see it will always lack the knowledge between good and evil and the choice of free will between them, that knowledge was imparted to man in the Garden of Eden by God, such knowledge is the subsequent reason why man's physicality must be remade to free us from sin forever by the resurrection.
The Christian Bible can explain science because it is the ultimate guide to understanding, but science will never explain the complete cosmos alone, it is hopelessly incapable but for the inspired Word of God Himself.