  We report on a sub-percent scale determination using the omega baryon mass
and gradient-flow methods. The calculations are performed on 22 ensembles of
$N_f=2+1+1$ highly improved, rooted staggered sea-quark configurations
generated by the MILC and CalLat Collaborations. The valence quark action used
is M\"obius Domain-Wall fermions solved on these configurations after a
gradient-flow smearing is applied with a flowtime of $t_{\rm gf}=1$ in lattice
units. The ensembles span four lattice spacings in the range $0.06 \lesssim a
\lesssim 0.15$ fm, six pion masses in the range $130 \lesssim m_\pi \lesssim
400$ MeV and multiple lattice volumes. On each ensemble, the gradient-flow
scales $t_0/a^2$ and $w_0/a$ and the omega baryon mass $a m_\Omega$ are
computed. The dimensionless product of these quantities is then extrapolated to
the continuum and infinite volume limits and interpolated to the physical
light, strange and charm quark mass point in the isospin limit, resulting in
the determination of $\sqrt{t_0}=0.1422(14)$ fm and $w_0 = 0.1709(11)$ fm with
all sources of statistical and systematic uncertainty accounted for. The
dominant uncertainty in this result is the stochastic uncertainty, providing a
clear path for a few-per-mille uncertainty, as recently obtained by the
Budapest-Marseille-Wuppertal Collaboration.
