Stealthiest configurations in two and three dimensions

Prove that the triangular lattice in two dimensions and the body-centered cubic lattice in three dimensions maximize stealthiness among particle configurations of fixed density.

Background

The paper establishes that the E8 and Leech lattices are the unique stealthiest isometry-invariant point processes of intensity 1 in dimensions 8 and 24. It contrasts these results with the conjectured optimizers in dimensions 2 and 3, where the analogous optimality statements remain unproved. The problem is to establish the corresponding global stealthiness bounds and identify the optimizers in those dimensions.