  The astrometry towards the Galactic Bulge is hampered by high stellar
crowding and patchy extinction. This effect is particularly severe for optical
surveys such as Gaia. In this study, we assess the consistency of proper
motions (PMs) between optical (Gaia DR3) and near-infrared (VIRAC2) catalogues
in comparison with PMs measured with the Hubble Space Telescope (HST) in
several crowded fields towards the Galactic Bulge and in Galactic globular
clusters. Assuming that the PMs are well characterised, the
uncertainty-normalised PM differences between pairs of catalogues are expected
to follow a normal distribution. A deviation from a normal distribution defines
the inflation factor $r$. Multiplying the PM uncertainties by $r$ brings the
Gaia (VIRAC2) PMs into a $1\sigma$ agreement with HST PMs. The factor $r$ has a
dependence on stellar surface density and for the brightest stars in our sample
(G<18), there is a strong dependence on G-band magnitude. Assuming that the HST
PMs are well determined and free from systematic errors, we find that Gaia DR3
PM uncertainties are better characterised, having r<1.5, in fields under 200
Gaia DR3 sources per arcmin$^2$, and are underestimated by up to a factor of 4
in fields with more than 300 Gaia DR3 sources per arcmin$^2$. For the most
crowded fields in VIRAC2, the PM uncertainties are underestimated by a factor
of 1.1 up to 1.5, with a dependence on J-band magnitude. In all fields, the
brighter sources have the larger $r$ value. At the faint end (G>19), $r$ is
close to 1, meaning that the PMs already fully agree with the HST measurements
within $1\sigma$. In the crowded fields with both catalogues in common, VIRAC2
PMs agree with HST PMs and do not need an inflation factor for their
uncertainties. Given the depth and completeness of VIRAC2 in such fields, it is
an ideal complement to Gaia DR3 for proper motion studies towards the Galactic
Bulge.
