Cohn–Kumar universal optimality conjecture for the hexagonal, E8, and Leech lattices

Let $n=2$, $8$, or $24$, and consider periodic configurations in $\mathbb{R}^n$. Their average potential energy per particle is defined using a completely monotonic potential function. The linear programming bounds are the Cohn–Kumar lower bounds for this energy obtained from suitable auxiliary functions. The hexagonal lattice, $E_8$, and the Leech lattice are the corresponding configurations in dimensions $2$, $8$, and $24$.

Cohn–Kumar conjecture. For $n=2$, $8$, or $24$, the linear programming bounds for potential energy minimization in $\mathbb{R}^n$ are sharp for every completely monotonic potential function (for the hexagonal, $E_8$, or Leech lattice, respectively).

This is a universal-optimality strengthening of optimality for sphere packings. The source states that even the two-dimensional case is not known, so the conjecture remains open.