  Measurements of neutral current Drell-Yan production at large invariant
dilepton masses can be used to test the energy scale dependence (running) of
the electroweak mixing angle. In this work, we make use of a novel
implementation of the full next-to-leading order electroweak radiative
corrections to the Drell-Yan process using the $\overline{\mathrm{MS}}$
renormalization scheme for the electroweak mixing angle. The potential of
future analyses using proton-proton collisions at $\sqrt{s}=13.6~\mathrm{TeV}$
in the Run 3 and High-Luminosity phases of the LHC is explored. In this way,
the Standard Model predictions for the $\overline{\mathrm{MS}}$ running at
$\mathrm{TeV}$ scales can be probed.
