The clique function's maximal flat-spectrum conjecture for the {I,H}^n transform set

Let $n$ be a positive integer, let $\{I,H\}^n$ denote the transform set consisting of $n$ choices between the identity transform $I$ and the Hadamard transform $H$, and let the clique function be the Boolean function defined in (clique). A Boolean function has a flat spectrum with respect to a transform set when all spectral coefficients have the same magnitude. Clique-function maximality conjecture. Over the set of all Boolean functions, the clique function maximizes the number of flat spectra with respect to $\{I,H\}^n$. Computational results establish this for $n\leq 5$ over all Boolean functions and for $n\leq 8$ over quadratic Boolean functions; the general claim remains open.