Universal optimality of the triangular lattice in the plane
In plain words
In 2022 the $E8$ and Leech lattices were proved to be the best arrangements in 8 and 24 dimensions for every repulsive force from a broad class. The matching claim for the triangular lattice in two dimensions is unproved.
Precise statement
Prove that for every density $\rho$ the triangular lattice minimizes the energy per point $\operatorname{sum}_{y != x} f(\mid x - y \mid^{2})$ among all periodic configurations in $R^{2}$ of density $\rho$, for every completely monotone f (including $\operatorname{exp}(-a r^{2})$ and $r^{-s}$). Answer: proof or a periodic configuration beating the triangular lattice for some such f.
What would settle it
A proof by an interpolation or linear-programming argument in the style of the 8D and 24D case, or an explicit counterexample.
Status in the literature
Proved for $E8$ and Leech (Cohn, Kumar, Miller, Radchenko, Viazovska, Annals 2022); in 2D known only among lattices; Petrache and Serfaty (2020) showed it would imply crystallization at next order for 2D Coulomb and Riesz gases.