  Choi, Kim, Kim, and Nahmgoong have recently pioneered analyzing a Cardy-like
limit of the superconformal index of the 4d $\mathcal{N}=4$ theory with
complexified fugacities which encodes the entropy of the dual supersymmetric
AdS$_5$ blackholes. Here we study the Cardy-like asymptotics of the index
within the rigorous framework of elliptic hypergeometric integrals, thereby
filling a gap in their derivation of the blackhole entropy function, finding a
new blackhole saddle-point, and demonstrating novel bifurcation phenomena in
the asymptotics of the index as a function of fugacity phases. We also comment
on the relevance of the supersymmetric Casimir energy to the blackhole entropy
function in the present context.
