  In the context of the Standard Model effective field theory (SMEFT) we
examine the constraints on the trilinear Higgs coupling that originate from
off-shell Higgs production in proton-proton collisions. Our calculation of the
$gg \to h^\ast \to Z Z \to 4 \ell$ process includes two-loop corrections to
gluon-gluon-fusion Higgs production and one-loop corrections to the Higgs
propagator and its decay. Employing a matrix-element based kinematic
discriminant we determine the reach of LHC Run 3 and the high-luminosity option
of the LHC in constraining the relevant SMEFT Wilson coefficients. We present
constraints that are not only competitive with but also complementary to the
projected indirect limits that one expects to obtain from inclusive
measurements of single-Higgs production processes at future LHC runs.
