  It was shown in a recent paper [J. Math. Phys. 60, 102502 (2019)] that slowly
lowering an electric charge into a Schwarzschild-Tangherlini (ST) black hole
endows the final state with electric multipole fields, which implies the final
state geometry is not Reissner-Nordstr\"om-Tangherlini in nature. This
conclusion departs from the four-dimensional case in which the no-hair theorem
(NHT) requires the final state to be a Reissner-Nordstr\"om black hole. To
better understand this discrepancy clearly requires a deeper understanding of
the origin of the multipole hair in the higher-dimensional case. In this paper,
we advance the conjecture that charged, static, and asymptotically-flat
higher-dimensional black holes can acquire electric multipole hair only after
they form. This supposition derives from studying the asymptotic behavior of
the field of a multipole charge onto which a massive and hyperspherical shell
with an exterior ST geometry is collapsing. In the mathematical limit as the
shell approaches its ST radius, we find that the multipole fields (except the
monopole) vanish. This implies that the only information of an arbitrary (but
finite) charge distribution inside the collapsing shell that is available to an
asymptotic observer is the total electric charge. Our results yield
considerable insight into how higher-dimensional black holes acquire electric
multipole hair, and also imply that, in four dimensions, the fadeaway of
multipole moments during gravitational collapse is not strictly because of the
NHT.
