  We evaluate the superconformal index of 4d $\mathcal{N}=4$ SYM with gauge
algebra $so(2N_c+1)$ in the Cardy-like limit. We then study the relation with
the results obtained for the S-dual $usp(2N_c)$, discussing the fate of
S-duality in different regions of charges. We find that S-duality is preserved
thanks to a non-trivial integral identity that relates the three sphere
partition functions of pure 3d Chern-Simons gauge theories.
