  Context. Most flybys in the Galactic disk are distant, beyond 10,000 AU, and
have characteristic velocities of about 70 km/s. However, deep and fast
encounters also take place, albeit with lower probability, particularly if one
of the objects involved is a stellar remnant ejected during a supernova. WD
0810-353 might be a high velocity white dwarf, and it was recently identified
as heading straight for the Solar System; however, the Gaia DR3 data that
support its future deep and fast flyby are regarded as suspicious.
  Aims. Here, we reanalyze the Gaia DR3 data set associated with WD 0810-353 to
confirm or reject the reality of its Solar System flyby and also to investigate
its possible runaway status.
  Methods. We studied the evolution of WD 0810-353 forward in time using N-body
simulations. We computed the distribution of distances of closest approach and
their associated times of perihelion passage. We used a statistical analysis of
the kinematics of this object to assess its possible hypervelocity. We compared
its mean BP/RP spectrum to those of other well-studied white dwarfs.
  Results. We confirm that WD 0810-353 is headed for the Solar System, but the
actual parameters of the encounter depend strongly on its radial velocity. The
Gaia DR3 value of -373.74+/-8.18 km/s is strongly disfavored by our analyses.
Its mean BP/RP spectrum suggests a value over ten times higher based on the
position of its putative Halpha line. However, spectral matching using other
white dwarfs with non-Gaia data indicate a radial velocity in the interval
(-60, -70) km/s.
  Conclusions. These results confirm the future flyby of WD 0810-353 near the
Solar System, although the relative velocity could be high enough or the
minimum approach distance large enough to preclude any significant perturbation
on the Oort cloud.
