The sharp zero-constant conjecture for the non-local interaction-energy bound

Let $C=1-\log 2<0.31$. For each finite set $\omega\subset\mathbb{R}\times(-\pi,\pi]$ with $|\omega|=N$, consider the lower bound

$$\sum_{z\neq z'\in\omega}W_\alpha(z-z')\geq-N(\log N+C).$$

Sharp zero-constant conjecture. Proposition~ should remain true with $C=0$; that is,

$$\sum_{z\neq z'\in\omega}W_\alpha(z-z')\geq-N\log N.$$

If true, this would imply that the minimum energy is attained by equally spaced points on a circle $x=x_0$. Numerical experiments support the conjecture, but no proof or resolution is given in the source.