  Gaia mission offers opportunities to search for compact binaries not involved
in binary interactions (hereafter inert compact binaries), and results in the
discoveries of binaries containing one black hole (BH) or one neutron star
(NS), called "Gaia BHs" and "Gaia NSs", respectively. We have assessed if Gaia
BHs and NSs can be formed in open clusters through dynamical interactions. In
order to obtain a large number of inert compact binaries similar to Gaia BHs
and NSs, we have performed gravitational $N$-body simulations for a large
number of open clusters whose total mass is $1.2 \times 10^8 M_\odot$. These
clusters have various masses, metallicities, densities, and binary fractions.
We have found that open clusters form Gaia BHs ($10^{-6}$-$10^{-5}
M_\odot^{-1}$) much more efficiently than Gaia NSs ($\lesssim 10^{-7}
M_\odot^{-1}$) for any cluster parameters. This is quite inconsistent with
observational results, because the reported numbers of Gaia BHs and NSs are $3$
and $21$, respectively. Additionally, we have switched off NS natal kicks for
$10^4$ open clusters each weighing $10^3 M_\odot$ in order to retain a large
number of NSs in open clusters. Then, open clusters form inert NS binaries
originating from primordial binaries rather than formed through dynamical
interactions. This means that Gaia NSs are formed dominantly on isolated
fields, not in open clusters, if there is no NS natal kick. We have concluded
that Gaia BHs can be dominantly formed in open clusters, however Gaia NSs
cannot.
