They replaced some of the lead(II) ions with copper(II), which has a smaller ionic radius. That's why the structure would be smaller. Then they meander around other superconductor discoveries to make the argument that this volume change is the cause of the superconducting property here.

As a chemist I can't say much about the physics aspects of this paper, but the chemistry and crystallography seem reasonable. It would only take a few days to replicate the synthesis and run some of these tests on it, assuming this is real. If I ran a lab doing these things I would be doing that right now.