They seem to be saying that any measurement regime, in achieving the necessary accuracy, will cause arbitrarily large measurement uncertainties. The authors at

They seem to be saying that any measurement regime, in achieving the necessary accuracy, will cause arbitrarily large measurement uncertainties. The authors at least, seem to think that this is fundamental, but as you say, it might just be a limitation of our current notion of a highly accurate clock. It might not matter though, if this is another in a long line of essentially non-falsifiable ideas.