Discreteness conjecture for non-even p-frame energy minimizers

Let $p>0$ with $p\notin 2\mathbb N$, and let $\mu$ be a probability measure on $\mathbb S^{d-1}$. Its $p$-frame energy is

$$I_f(\mu)=\int_{\mathbb S^{d-1}}\int_{\mathbb S^{d-1}}|\langle x,y\rangle|^p\,d\mu(x)d\mu(y).$$

A finite discrete measure is a measure supported on finitely many points. Discreteness conjecture. Every minimizer of $I_f$ is a finite discrete measure on $\mathbb S^{d-1}$. For even integer $p$, continuous and discrete minimizers are plentiful, whereas for non-even $p$ the asserted discreteness is supported by results for tight designs and numerical evidence; the conjecture remains open in general.