First off, I want to emphasize my model of with a convex space was merely illustrative, to build some intuition here. It may break down in much higher dimensional space.
So here is the statement in conflict with the data…
Our test is based on the expectation that, under evolution, the ancestral sequence of one natural group of taxa will be more similar to the ancestral sequence of a second natural group of taxa, than to any sequence from the first group will be to any sequence from the second.
They talk about two things here
A. The difference between two inferred ancestral sequences of two taxa.
B. the difference between two sequences, each from a different one of the two taxa.
They say that A will be smaller than B for any pair of sequences from the two taxa. That any is where the problem is. Look at these data figures sliced from Figure 3:
The circle is where the ancestral sequence pair is. The line shows the distribution of inter-tax sequence pairs. Note, that in ALL cases, there exists at least a few examples of sequence pairs that have higher similarity than the ancestral sequence pair. Moreover, for some cases (psbL) there ancestral sequence is less similar than the bulk of sequence pairs.
Therefore, it is not true that any inter-taxa sequence pair is lower similarity than the ancestral sequence pair. If we take their sentence at face value, this should be evidence against common descent. Really, it is just sloppy language, and perhaps another error is affecting them too (lack of controls).
The next sentence says:
In contrast, a variety of proposed non-evolutionary models either do not make this prediction, or require so many parameters that they cannot be said to make any testable predictions at all.
It is not clear what these non-evolutionary models are to which the authors refer. I’ve proposed one that I think will produce similar results. Maybe I am wrong, but one cannot demonstrate that all possible special creation models do not make this prediction. We can make claims, however, about known models, but we actually have to test what we claim are predictions, ideally with simulation. They just assert the models don’t make this prediction, but it is not clear they have demonstrated it to be so, or even identified what these models actually are.
This gets to the final problem. They did not demonstrate that common descent makes this prediction. It is important to remember that population genetics is not intuitive. It is critical to test claims with simulation. They did not do this. Instead, the just asserted what the common descent expectation is, without ever testing to see if this is actually correct. That is, perhaps, why their is sloppy language. They weren’t actually testing their claims about what each model predicts.
So, their claims about what common descent predicts ends up actually contradicting the evidence they present. That is not a good situation to be in.