  We implemented combination of DMFT and $GW$ in its fully self-consistent way,
one shot $GW$ approximation, and quasiparticle self-consistent scheme, and
studied how well these combined methods perform on H$_2$ molecule as compared
to more established methods such as LDA+DMFT. We found that most flavors of
$GW$+DMFT break down in strongly correlated regime due to causality violation.
Among $GW$+DMFT methods, only the self-consistent quasiparticle $GW$+DMFT with
static double-counting, and a new method with causal double-counting, correctly
recover the atomic limit at large H-atom separation. While some flavors of
$GW$+DMFT improve the single-electron spectra as compared to LDA+DMFT, the
total energy is best predicted by LDA+DMFT, for which the exact double-counting
is known, and is static.
