What does this tell us about the evolution of competence in particular, and the evolution of H. influenzae in general? Certainly the alleles of competence genes are mixed between strains (I also have evidence supporting this), an observation consistent with the phenotypic variation we observe in DNA uptake and transformation. This tells us that the ability to take up DNA and recombine it changes relatively quickly during the evolution of strains. This also tells us that strains of H. influenzae are not diverging from one another but that they form a cohesive group, linked by the sharing of alleles.
But even though my inability to infer a resolved tree tells us interesting things about the evolution of competence and H. influenzae, it is still unsatisfying. And I keep asking myself what other methods or datasets could be used. But I have to stop somewhere and move on from here.