Universal optimality of the hexagonal, $E_8$, and Leech lattices for charged periodic configurations

Let $f:(0,\a∞)\to\mathbb{R}_+$ be completely monotone and satisfy $|f(r)|=O(r^{-d/2-\varepsilon})$ as $r\to+\infty$ for some $\varepsilon>0$. For $N\in2\mathbb{N}$, let $\mathcal{A}_N^1$ be the space of unit-density periodic configurations with $N$ points per period, and let $\Phi_N(\mathcal{C})$ be the neutral periodic charge distributions with charges in $\{-1,1\}$. Define

$$E_f^\pm[\mathcal{C}]:=\min_{\varphi\in\Phi_N(\mathcal{C})}E_f[\mathcal{C},\varphi],$$

where

$$E_f[\mathcal{C},\varphi]:=\frac{1}{N}\sum_{i=1}^N\sum_{q\in\mathcal{C}\setminus\{t_i\}}\varphi(t_i)\varphi(q)f(|t_i-q|^2).$$

Universal optimality conjecture. For all $N\in2\mathbb{N}$, $\mathsf{A}_2$, $\mathsf{E}_8$, and the Leech lattice are the unique maximizers of $E_f^\pm$ on $\mathcal{A}_N^1$ in the respective dimensions $d\in\{2,8,24\}$. This asserts universal optimality among periodic configurations with charges $\pm1$.