Fairness in access to HE is unarguably a subject of paramount importance. Wherever a student's secondary school scores are relevant for access to HE, grade inflation practices may jeopardize fair access. Pressures for high grading are common in the context of educational consumerism and competition between schools and students. However, they are not equally distributed across different types of schools, given that they have distinct relationships with the State and the market, and work with distinct populations. Specifically, the schools that are more subject to market pressures (namely private schools) are, in principle at least, the ones with more incentives to inflate their students' grades. This paper presents an empirical study based on a large, 11 years database on scores in upper secondary education in Portugal, probing for systematic differences in grade inflation practices by four types of schools: public schools, government-dependent private schools, independent (fee-paying) private schools, and specially funded public schools in disadvantaged areas (TEIP schools). More than 3 million valid cases were analysed. Our results clearly show that independent private schools inflate their students' scores when compared to the other types of schools. They also show that this discrepancy is higher where scores matter most in competition for HE access. This means that--usually wealthier--students from private independent schools benefit from an unfair advantage in the competition for the scarce places available in public higher education. We conclude discussing possible solutions to deal with such an important issue.