Dynamical History Of The Local Group In $\Lambda$CDM $\rm{II}$ $-$
  Including External Perturbers In 3D
  We attempt to fit the observed radial velocities (RVs) of ${\,{\sim}\,}30$
Local Group (LG) galaxies using a 3D dynamical model of it and its immediate
environment within the context of the standard cosmological paradigm,
$\Lambda$CDM. This extends and confirms the basic results of our previous
axisymmetric investigation of the LG (MNRAS, 459, 2237). We find that there
remains a tendency for observed RVs to exceed those predicted by our
best-fitting model. The typical mismatch is slightly higher than in our 2D
model, with a root mean square value of ${\,{\sim}\,}50$ km/s. Our main finding
is that including the 3D distribution of massive perturbing dark matter halos
is unlikely to help greatly with the high velocity galaxy problem. Nonetheless,
the 2D and 3D results differ in several other ways such as which galaxies' RVs
are most problematic and the preferred values of parameters common to both
models.
  The anomalously high RVs of several LG dwarfs may be better explained if the
Milky Way (MW) and Andromeda (M31) were once moving much faster than in our
models. This would allow LG dwarfs to gain very high RVs via gravitational
slingshot encounters with a massive fast-moving galaxy. Such a scenario is
possible in some modified gravity theories, especially those which require the
MW and M31 to have previously undergone a close flyby. In a $\Lambda$CDM
context, however, this scenario is not feasible as the resulting dynamical
friction would cause a rapid merger.
