Equality between the radius-two lattice neighborhood number and the lattice kissing number

Determine whether τ*₂(Bⁿ) = κ*(Bⁿ) + 1 holds for every n-dimensional unit ball Bⁿ, where τ*₂(Bⁿ) is the maximum number of lattice points within distance 2 of a point in a lattice packing and κ*(Bⁿ) is the lattice kissing number.

Background

The paper introduces τρ(C) as the largest number of lattice points that can lie within C-distance ρ of a point, over all lattice packings of a centrally symmetric convex body C. General inequalities relate τρ(C) to generalized lattice kissing numbers and to lattice packing density.

For arbitrary centrally symmetric convex bodies, the corresponding translative quantity satisfies τ₂(C) = κ(C) + 1. The authors ask whether the analogous equality holds for lattice packings of Euclidean balls. They note that a positive answer would imply a density bound involving κ*(Bⁿ), while a negative answer would itself yield an interesting counterexample.