Extend Theorem 1.2 to all origin-symmetric convex bodies of volume c n^2

Establish that for every dimension n ≥ 2 and every origin-symmetric convex body K ⊂ ℝ^n with Vol_n(K) = c n^2 for some universal constant c > 0, there exists a full-rank lattice L ⊂ ℝ^n of covolume one such that L ∩ K = {0}.

Background

Theorem 1.2 of the paper proves the existence of a lattice L of covolume one that avoids the origin-centered Euclidean ball K of volume c n^2 (i.e., L ∩ K = {0}). The authors then observe that this implies the existence of an origin-symmetric ellipsoid of volume c n^2 containing no nonzero points of ℤ^n.

They conjecture that this lattice-avoidance conclusion should hold for any origin-symmetric convex body K satisfying the same volume constraint Vol_n(K) = c n^2, not just for Euclidean balls or ellipsoids. Schmidt previously established a related statement under a stronger small-volume assumption Vol_n(K) ≤ c n, leaving the c n^2 threshold open in the general convex-body setting.