Universal optimality of the triangular lattice in the plane

Prove that for every density $\rho$ the triangular lattice minimizes the energy per point $\sum_{y != x} f(\mid x - y \mid^{2})$ among all periodic configurations in $R^{2}$ of density $\rho$, for every completely monotone f (including $\exp(-a r^{2})$ and $r^{-s}$). Answer: proof or a periodic configuration beating the triangular lattice for some such f.

A proof by an interpolation or linear-programming argument in the style of the 8D and 24D case, or an explicit counterexample.