  A higher-order topological insulator (HOTI) in two dimensions is an insulator
without metallic edge states but with robust zero-dimensional topological
boundary modes localized at its corners. Yet, these corner modes do not carry a
clear signature of their topology as they lack the anomalous nature of helical
or chiral boundary states. Here, we demonstrate using immunity tests that the
corner modes found in the breathing kagome lattice represent a prime example of
a mistaken identity. Contrary to previous theoretical and experimental claims,
we show that these corner modes are inherently fragile: the kagome lattice does
not realize a higher-order topological insulator. We support this finding by
introducing a criterion based on a corner charge-mode correspondence for the
presence of topological midgap corner modes in n-fold rotational symmetric
chiral insulators that explicitly precludes the existence of a HOTI protected
by a threefold rotational symmetry.
