Optimality conjecture for the harmonic kernel in Riesz energy

Let $cmathbb{S}^d$ be the unit sphere and let $n=cpi_L$. Let $K$ be an isotropic kernel producing $n$ points through its determinantal point process, and let $E_s(x)$ denote the Riesz $s$-energy of a configuration $x$. The harmonic ensemble is the determinantal process associated with the harmonic kernel. Harmonic-kernel optimality conjecture. For every $sgeq0$, the expected energy for any such kernel $K$ is at least the expected energy for the harmonic ensemble:

$$cmathbb{E}_{K}(E_s(x))\geq cmathbb{E}_{ctext{harmonic}}(E_s(x)).$$

The preceding theorem proves an analogous optimality statement for $s=2$ under specified coefficient conditions; the conjecture asks whether harmonic-kernel optimality persists for all $sgeq0$.