  The Oktay-Kronfeld (OK) action extends the Fermilab improvement program for
massive Wilson fermions to higher order in suitable power-counting schemes. It
includes dimension-six and -seven operators necessary for matching to QCD
through order ${\mathrm{O}}(\Lambda^3/m_Q^3)$ in HQET power counting, for
applications to heavy-light systems, and ${\mathrm{O}}(v^6)$ in NRQCD power
counting, for applications to quarkonia. In the Symanzik power counting of
lattice gauge theory near the continuum limit, the OK action includes all
${\mathrm{O}}(a^2)$ and some ${\mathrm{O}}(a^3)$ terms. To assess whether the
theoretical improvement is realized in practice, we study combinations of
heavy-strange and quarkonia masses and mass splittings, designed to isolate
heavy-quark discretization effects. We find that, with one exception, the
results obtained with the tree-level-matched OK action are significantly closer
to the continuum limit than the results obtained with the Fermilab action. The
exception is the hyperfine splitting of the bottom-strange system, for which
our statistical errors are too large to draw a firm conclusion. These studies
are carried out with data generated with the tadpole-improved Fermilab and OK
actions on 500 gauge configurations from one of MILC's $a\approx0.12$~fm,
$N_f=2+1$-flavor, asqtad-staggered ensembles.
