Isoperimetric conjecture for the alpha-Riesz energy

Let $N\geq 2$, let $\alpha\in(0,N)$, and let $I_{\alpha}(K)$ denote the $\alpha$-Riesz energy of a convex body $K\subset\mathbb{R}^N$. For $r>0$, consider convex bodies with perimeter $P(K)=N\omega_N r^{N-1}$. Isoperimetric conjecture for the $\alpha$-Riesz energy. The ball of radius $r$ is the unique solution of

$$\min_{K\in\mathcal{K}_N,\,P(K)=N\omega_N r^{N-1}} I_{\alpha}(K).$$

The paper has established the corresponding minimization result for the $1$-Riesz energy under fixed mean width, including the planar perimeter formulation; this conjecture asks for uniqueness for every $\alpha\in(0,N)$ under fixed perimeter.