Bilyk–Glazyrin–Matzke–Park–Vlasiuk's discreteness conjecture for p-frame minimizers

Let $p>0$, and for a probability measure $\mu$ on $\mathbb{S}^{d-1}$ define the $p$-frame energy by

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

A minimizer is a probability measure attaining the minimum of this energy. Bilyk–Glazyrin–Matzke–Park–Vlasiuk's discreteness conjecture. For $p>2$ and $p\neq 2k$, $k\in\mathbb{N}$, all minimizers of the $p$-frame energy are discrete measures. The question concerns the structure of minimizers for the values of $p$ not covered by the known even-integer cases. The supplied text gives no evidence that this statement has been resolved.