  Bekenstein's generalized second law (GSL) of thermodynamics asserts that the
sum of black-hole entropy, $S_{\text{BH}}=Ac^3/4\hbar G$ (here $A$ is the
black-hole surface area), and the ordinary entropy of matter and radiation
fields in the black-hole exterior region never decreases. We here re-analyze an
intriguing gedanken experiment which was designed by Bekenstein to challenge
the GSL. In this historical gedanken experiment an entropy-bearing box is
lowered into a charged Reissner-Nordstr\"om black hole. For the GSL to work,
the resulting increase in the black-hole surface area (entropy) must compensate
for the loss of the box's entropy. We show that if the box can be lowered
adiabatically all the way down to the black-hole horizon, as previously assumed
in the literature, then for near-extremal black holes the resulting increase in
black-hole surface-area (due to the assimilation of the box by the black hole)
may become too small to compensate for the loss of the box's entropy. In order
to resolve this apparent violation of the GSL, we here suggest to use a
generalized version of the hoop conjecture. In particular, assuming that a
physical system of mass $M$ and electric charge $Q$ forms a black hole if its
circumference radius $r_{\text{c}}$ is equal to (or smaller than) the
corresponding Reissner-Nordstr\"om black-hole radius
$r_{\text{RN}}=M+\sqrt{M^2-Q^2}$, we prove that a new (and larger) horizon is
already formed before the entropy-bearing box reaches the horizon of the
original near-extremal black hole. This result, which seems to have been
overlooked in previous analyzes of the composed black-hole-box system, ensures
the validity of Bekenstein's GSL in this famous gedanken experiment.
