It's worth nothing that there are a number of limitations to the study, some of which are acknowledged in the paper itself.
Firstly, the study only measures the results of science, maths and English GCSEs. Those are the subjects that are most likely to be set in state schools, because they have the largest numbers of pupils, and because maths in particular is perceived as a hierarchical subject. So the potential influence of a lesser degree of setting in subjects like MFL or history in state schools won't appear in the analysis.
The authors also dont acknowledge the potential ceiling effect that arises from using GCSEs as a measure, and especially those particular GCSEs. (One of the really odd omissions here is that they refer to an "English" GCSE, but don't seem to specify which one - even though they explain about triple science, and that they used an average of the 3 scores for that.) Especially if it is English Language they chose to use, the ceiling effect in that particular exam would have a big impact.
By "ceiling effect" I mean that GCSEs don't measure broader learning "above and beyond" what is required to get an A* at GCSE: learning that will be especially useful for A levels and university, as well as worthwhile in itself . This type of learning is much more significant for those whose A levels and degrees will be in humanities subjects, IMO , which is why I am concerned about the narrow range of GCSEs covered in the study.
Still, it's an interesting paper, even though I don't think it can be regarded as definitive. I'd be interested in the views of scientists/ mathematicians/psychologists on the methods of analysis and the reliability of the adjustments they made to the raw data.
You can read the whole thing here: www.nature.com/articles/s41539-018-0019-8