A classical school choice problem consists of a set of schools with priorities over students and a set of students with preferences over schools. Schools’ priorities are often based on multiple criteria, for example, merit-based test scores as well as minimal-access rights (siblings attending the school, students’ proximity to the school, etc.). Traditionally, minimal-access rights are incorporated into priorities by always giving minimal-access students higher priority over non-minimal-access students. However, stability based on such adjusted priorities can be considered unfair because a minimal-access student may be admitted to a popular school, whereas another student with a higher merit score but without a minimal-access right is rejected, even though the former minimal-access student could easily attend another of her minimal-access schools. We therefore weaken stability to minimal-access stability: minimal-access rights promote access to only at most one minimal-access school. Apart from minimal-access stability, we also would want a school choice mechanism to satisfy strategy-proofness and minimal-access monotonicity, that is, additional minimal-access rights for a student do not harm her. Our main result is that the deferred acceptance mechanism is the only mechanism that satisfies minimal-access stability, strategy-proofness, and minimal-access monotonicity. Because this mechanism is in fact stable, our result can be interpreted as an impossibility result: fairer outcomes that are made possible by the weaker property of minimal-access stability are incompatible with strategy-proofness and minimal-access monotonicity. Funding: This work was supported by the Spanish State Research Agency [Grant PID2020-114251GB-I00], AGAUR–Generalitat de Catalunya [Grant 2021-SGR-00416], the Severo Ochoa Programme for Centres of Excellence in R&D (Barcelona School of Economics) [Grant CEX2019-000915-S], and the Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung [Grant 100018_192583].