Universal optimality of the hexagonal, E8, and Leech lattices among periodic configurations

Let $\Lambda_2$, $\Lambda_8$, and $\Lambda_{24}$ be the hexagonal lattice in $\mathbb{R}^2$, the $E_8$ root lattice in $\mathbb{R}^8$, and the Leech lattice in $\mathbb{R}^{24}$, respectively. Let $f\colon(0,\infty)\to\mathbb{R}$ be completely monotonic and satisfy

$$f(x)=O\bigl(|x|^{-n/2-\varepsilon}\bigr)\quad\text{as }|x|\to\infty$$

for some $\varepsilon>0$. A periodic configuration has least $f$-potential energy at its density if its $f$-potential energy is no greater than that of every periodic configuration of the same density. Universal optimality conjecture. For $n\in\{2,8,24\}$, there exists a function $h$ satisfying the hypotheses of Proposition 10.1, or the proposition referred to in the source, such that $h$ proves that $\Lambda_n$ has the least $f$-potential energy of any periodic configuration in $\mathbb{R}^n$ with its density. If true, these lattices would be universally optimal configurations in Euclidean space; the statement remains open in the source.