We investigate the impact of flavor-conserving, non-universal quark-lepton contact interactions on the dilepton invariant mass distribution in p p → + − processes at the LHC. After recasting the recent ATLAS search performed at 13 TeV with 36.1 fb −1 of data, we derive the best up-to-date limits on the full set of 36 chirality-conserving four-fermion operators contributing to the processes and estimate the sensitivity achievable at the HL-LHC. We discuss how these high-p T measurements can provide complementary information to the low-p T rare meson decays. In particular, we find that the recent hints on lepton-flavor universality violation in b → sμ + μ − transitions are already in mild tension with the dimuon spectrum at high-p T if the flavor structure follows minimal flavor violation. Even if the mass scale of new physics is well beyond the kinematical reach for on-shell production, the signal in the high-p T dilepton tail might still be observed, a fact that has been often overlooked in the present literature. In scenarios where new physics couples predominantly to third generation quarks, instead, the HL-LHC phase is necessary in order to provide valuable information.