  We investigate a simple-sum giant graviton expansion of the superconformal
indices of ${\cal N}=2$ superconformal field theories realized on D3-branes
probing $7$-brane backgrounds with constant axio-dilation field. The expansion
is of self-dual type, and imposes strong constraints on the indices. By using
the constraints we determine first few terms in the superconformal indices for
arbitrary rank $N$.
