Discreteness of minimizers for p-frame energy when p>2 and p is not an even integer

Determine whether, for the p-frame energy I_F(μ) = ∫_{S^{d−1}}∫_{S^{d−1}} |⟨x, y⟩|^p dμ(x) dμ(y) on the unit sphere S^{d−1}, every minimizer is a discrete measure whenever p > 2 and p is not an even integer.

Background

The p-frame energy functional integrates the pth power of absolute inner products over probability measures on the sphere. Known results identify minimizers for special p: tight frames and isotropic measures for p = 2, orthonormal bases for 0 < p < 2, and spherical designs/uniform measure for all even p > 2.

For other p, numerical experiments suggest minimizers are discrete, and supports of minimizers have empty interior if p is not an even integer. The conjecture seeks to establish discreteness of all minimizers under p > 2 and p ∉ 2ℕ.