Correct me if I'm misreading, but I have a hard time seeing how the study supports your hypothesis. In the "Discussion" section, the authors state that their heritability estimate for educational attainment was 31%, and only 20% for SES. That still leaves the door open for significant social influence.

The authors summarize: "Our findings add weight to the view that genetic variation plays an important, but not exclusive, role in educational inequalities and social mobility, which is at variance with views, that still prevail in some quarters, that these are solely the product of social forces and environmental inequalities." In other words, they only take themselves to have refuted the strawman view that observed differences "are solely the product of social forces."