  We investigate the impact of flavour-conserving, non-universal quark-lepton
contact interactions on the dilepton invariant mass distribution in $p~p \to
\ell^+ \ell^-$ 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 relevant four-fermion operators, as well as
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
flavour universality violation in $b \to s \mu^+ \mu^-$ transitions are already
in mild tension with the dimuon spectrum at high-$p_T$ if the flavour structure
follows minimal flavour 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.
