No-local-minimum conjecture for the discrete signed-mass energy

Let $x=(x_1,\ldots,x_d)$ with $x_i\in{\mathbb R}^n$, signed masses $m(x_i)$ satisfying

$$\sum_{i=1}^d |m(x_i)|^2=1,$$

and define

$$E(m(x))=\sum_{i,j=1}^d\frac{-2m(x_i)m(x_j)}{|x_i-x_j|^{n+2s}}.$$

A configuration is stationary and stable when its first variation vanishes and its symmetric second variation is nonnegative in every direction. No-local-minimum conjecture. The energy $E$ has no local minimum for $d\geq 2$. This is a discrete toy analogue of the proposed behavior for interacting components of fractional-Laplacian eigenvalue minimizers; the source introduces it as an associated conjecture and gives no resolution.