I pretty much have to take the anatomical data, and the egyptological discussion, on faith - I have insufficient knowledge in either of those fields to examine the data critically. The genetic analysis on the other hand is closer to my own field, so I can poke at it with a more professional & critical eye. Though you should bear in mind that I'm 5 years out of date and this wasn't quite my field when I was current. So before I go through the actual data, here's some thoughts on the methods/presentation.
They don't discuss the details of the extraction of the DNA from the bones of the mummies - they do however cite 4 papers (only 2 of which have overlapping authors with this paper) for the method they use, so I'm inclined to think (without reading those papers) that this is a known and tried method in the field of ancient DNA extraction.
They use commercially available kits to do the genetic fingerprinting, from a quick glance through the company's website they're used for more modern forensic applications too. I was, though, concerned that although they use 16 markers on the Y-chromosome analysis they only use 8 markers for the autosomal DNA (the rest of the DNA in the nucleus that is not sex chromosomes). The kit seemed to allow you to test more markers than that - but they don't mention why only 8. Was it that they had good data for those 8? or is this an accepted practice? (as I said, this isn't quite my field). Some note of that might have been nice.
They did analysis on 30 samples for each mummy, taken from different biopsies - it might've been nice to have seen a figure with the biopsy locations marked on it (in the supplemental data perhaps). And it would've been nice to see the numbers - the data given is noted as the "majority", but I think I would've liked to see the actual figures. It makes a difference if something came up in 16 of the samples or in 29 of the samples.
I would also have liked to see some controls! In the Y-chromosome analysis they do compare the three male mummies with another unrelated one. But in the bulk of the data they don't show the control data.