Not to mention the tricky question of what happens to your consciousness during and after this process?

Most likely they're just preserving the tissue, but not the consciousness

Isn't consciousness more like what emerges out of the neurons firing? If I turn off my calculator, I can't input calculations into it and get an answer, but if I turn it on, the electricity running in the circuitry will react to the button presses, and I can say "the calculator is working".

So the consciousness is the "computing session", and if we can persist the state of the hardware before shutdown, then booting it back up will give us that consciousness.