  A few years ago it was shown that the superconformal index of the
$\mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory in the large $N$ limit
matches with the entropy of $1/16$-supersymmetric black holes in type IIB
string theory on $AdS_5\times S^5$. In some cases, an even more detailed match
between the two sides is possible. When the two angular momentum chemical
potentials in the index are equal, the superconformal index can be written as a
discrete sum of Bethe ansatz solutions, and it was shown that specific terms in
this sum are in a one-to-one correspondence to stable black hole solutions, and
that the matching can be extended to non-perturbative contributions from
wrapped D3-branes. A Bethe ansatz approach to computing the superconformal
index exists also when the ratio of the angular momentum chemical potentials is
any rational number, but in those cases it involves a sum over a very large
number of terms (growing exponentially with $N$). Benini et al showed that a
specific one of these terms matches with the black hole, but the role of the
other terms is not clear. In this paper we analyze some of the additional
contributions to the index in the Bethe ansatz approach, and we find that their
matching to the gravity side is much more complicated than in the case of equal
chemical potentials. In particular, we find some contributions that are larger
than the one which was found to match the black holes, so that they must cancel
with other large contributions. We give some evidence that cancellations of
this type are possible, but we leave a full understanding of how they work to
the future.
