I have only skimmed the paper, but have already seen that the author does not claim, and in fact disclaims, that the paper undermines common descent. He admits that the paper ignores multiple known facts of evolutionary genetics and natural history, but insists that this should not hinder consideration of his model. To some extent, I agree. Here is the text I refer to, from page 18 of the paper:
An obvious objection is that we have not included any of the mechanisms thought to account for nonhierarchical data such as incomplete lineage sorting, gene flow, convergent evolution, or horizontal gene transfer. As such, it might be argued that any of the features of the data interpreted as evidence for the dependency graph may also be explained by these mechanisms. The focus of this paper has not been to critique common descent, but to the test the predictions of the dependency graph hypothesis. The challenge to common descent lies not in the comparison of the tree and dependency graph models but in explaining the successful predictions of the dependency graph hypothesis.
What I found a lot less clear was the reasoning or data behind some of the specific scientific claims. Here is the one I am most interested in, from the beginning of the section “Small examples” on page 12:
We will now consider a few small examples of cases which our model-fitting method inferred to be explained by modules. A striking example can be found in Nematostella vectensis (scarlet sea anemone) and Branchiostoma floridae (Florida lancelet). These are distantly related organisms with the anemone being in the phylum Cnidaria and the lancelet being in the phylum Chordata. Nevertheless, they contain between 25 and 564 (depending on the database consulted) gene families found in both species but in no other metazoan species in the database. In all datasets where both species are present, a module is inferred to exist to explain the genes found in both species.
I strongly suspect that this claim is an uninteresting artifact of how the author is interpreting a database, and specifically how he is building conclusions from the fact that something is “not found” somewhere, but if there is any basis to it at all, it would be very interesting. Does anyone know, or can you tell, where he even got this? I’ll look harder too.