  Both major galaxies in the Local Group host planar distributions of
co-orbiting satellite galaxies, the Vast Polar Structure (VPOS) of the Milky
Way and the Great Plane of Andromeda (GPoA). The $\Lambda$CDM cosmological
model did not predict these features. However, according to three recent
studies the properties of the GPoA and the flattening of the VPOS are common
features among sub-halo based $\Lambda$CDM satellite systems, and the GPoA can
be naturally explained by satellites being acquired along cold gas streams. We
point out some methodological issues in these studies: either the selection of
model satellites is different from that of the observed ones, or an incomplete
set of observational constraints has been considered, or the observed satellite
distribution is inconsistent with basic assumptions. Once these issues have
been addressed, the conclusions are different: features like the VPOS and GPoA
are very rare (each with probability $\lesssim 10^{-3}$, and combined
probability $< 10^{-5}$) if satellites are selected from a $\Lambda$CDM
simulation combined with semi-analytic modelling, and accretion along cold
streams is no natural explanation of the GPoA. The origin of planar dwarf
galaxy structures remains unexplained in the standard paradigm of galaxy
formation.
