Diet is not expected to have any influence of parent relations, social relations or peers but your study may find an influence on psychological well-being and even school environment.
The conclusion is sound given the findings but the lack of reference to income is an issue as educational status of parents is often a “de-facto” measure for income. It maybe, it is the cost of living associated with fresh fruit and vegetables that is the real issue. Given the findings the researchers could be arguing that a review may be needed with the Swedish school lunch program and increase the level of fruit and vegetables in the students' diet, particularly for students in lower SES locations. Schools may be making the lunches to a budget rather than to a healthy diet criteria which is more expensive. Different sub-populations even in the same school may need different mix of foods. A public education program could also be encouraged about health eating.