Uniqueness of the diffusively dominant circle packing

A circle packing is a packing of congruent circles in the plane, and a packing is diffusively dominant if its mass distribution diffusively dominates that of every other packing. Circle-packing conjecture. The hexagonal packing is the unique diffusively dominant circle packing. This is a proposed uniqueness principle for circle packings under diffusive dominance; the supplied text gives no resolution of the conjecture.