Dara, I simply make the comment that the paper is reporting this attribution. I have done the correlations myself and straightforwardly get correlation coefficients that can reach 0.7 or higher as shown above in #9, which is nothing to sneeze at.
I realize that research papers can be frustrating because all the methods and computations are not immediately visible, but that is the way it has always been and so I am used to it by now. I place some trust that the researchers aren't leading us astray. (I can tell you "research" that I won't trust, if you are interested)
The next part is to figure out what part of the local sea-level height change is due to (1) sloshing of the volume (2) inverse barometric effect (3) thermal expansion of upwelling water and (4) changes in precipitation. The last one can have severe impacts for Australia, in particular, as the Australian great artesian basin is an excellent transient trap for excess precipitation which can then depress ocean levels temporarily