  We recently showed that several Local Group (LG) galaxies have much higher
radial velocities (RVs) than predicted by a 3D dynamical model of the standard
cosmological paradigm. Here, we show that 6 of these 7 galaxies define a thin
plane with root mean square thickness of only 101 kpc despite a widest extent
of nearly 3 Mpc, much larger than the conventional virial radius of the Milky
Way (MW) or M31. This plane passes within ${\sim 70}$ kpc of the MW-M31
barycentre and is oriented so the MW-M31 line is inclined by $16^\circ$ to it.
  We develop a toy model to constrain the scenario whereby a past MW-M31 flyby
in Modified Newtonian Dynamics (MOND) forms tidal dwarf galaxies that settle
into the recently discovered planes of satellites around the MW and M31. The
scenario is viable only for a particular MW-M31 orbital plane. This roughly
coincides with the plane of LG dwarfs with anomalously high RVs.
  Using a restricted $N$-body simulation of the LG in MOND, we show how the
once fast-moving MW and M31 gravitationally slingshot test particles outwards
at high speeds. The most distant such particles preferentially lie within the
MW-M31 orbital plane, probably because the particles ending up with the highest
RVs are those flung out almost parallel to the motion of the perturber. This
suggests a dynamical reason for our finding of a similar trend in the real LG,
something not easily explained as a chance alignment of galaxies with an
isotropic or mildly flattened distribution (probability $= {0.0015}$).
