It is well known, at least under the presence of inequality constraints, that strong first order constraint qualifications like Mangasarian-Fromovitz do not ensure the validity of a second order necessary condition. Recent works try to remedy this situation using constant rank assumptions that are weaker than linear independence of all active gradients. Such assumptions usually involve the gradients of all subsets of active inequality constraints. In this work we discuss the role of such constant rank conditions, showing that they can be greatly relaxed even in the presence of inequalities. In particular, it is possible to select a single subset of active inequalities whose good behavior together with all equalities is enough to show that a necessary second order condition holds at a local minimum for any given multiplier.