I think it is generally unwise to assume that a single genetic data set can provide a definitive resolution of any issue. To be really confident about something, I need some type of data corroboration, preferably from an independent source, and this is cruelly lacking here. But as it stands, I would say that the part of the data set that is not obviously incorrect appears to contradict the relationships suggested in the article...
I also do not believe that a tree can still be trusted if the data set is shown to have included chimeric sequences--eg., if some terminals are actually represented by a mixture of sequences from different taxa, like here a Rostratula ND2 concatenated to Recurvirostra Cytochrome b, 12S-rRNA and RAG-1. Such chimeras create attractions within the data set that should not have been there, and this affects the whole tree reconstruction, not just their own position. If shown to be present, they must be removed, and the analysis must be entirely re-done.
Unfortunately, there are many such sequences in this particular data set.