  The large $N$ limit of the four-dimensional superconformal index was computed
and successfully compared to the entropy of a class of AdS$_5$ black holes only
in the particular case of equal angular momenta. Using the Bethe Ansatz
formulation, we compute the index at large $N$ with arbitrary chemical
potentials for all charges and angular momenta, for general $\mathcal{N}=1$
four-dimensional conformal theories with a holographic dual. We conjecture and
bring some evidence that a particular universal contribution to the sum over
Bethe vacua dominates the index at large $N$. For $\mathcal{N}=4$ SYM, this
contribution correctly leads to the entropy of BPS Kerr-Newman black holes in
AdS$_5 \times S^5$ for arbitrary values of the conserved charges, thus
completing the microscopic derivation of their microstates. We also consider
theories dual to AdS$_5 \times \mathrm{SE}_5$, where SE$_5$ is a
Sasaki-Einstein manifold. We first check our results against the so-called
universal black hole. We then explicitly construct the near-horizon geometry of
BPS Kerr-Newman black holes in AdS$_5 \times T^{1,1}$, charged under the
baryonic symmetry of the conifold theory and with equal angular momenta. We
compute the entropy of these black holes using the attractor mechanism and find
complete agreement with the field theory predictions.
