D4 lattice optimal sphere packing in four dimensions

Prove that the D4 lattice achieves the optimal sphere packing density in four-dimensional Euclidean space R^4 among all sphere packings (not restricted to lattice packings).

Background

The authors review the state of the sphere packing problem: Viazovska and collaborators resolved the optimal sphere packing in dimensions 8 and 24 via sharp Cohn–Elkies bounds, while dimension 4 remains unresolved. Although optimality among lattice packings in dimension four has been known since Korkine and Zolotareff (1873), proving global optimality among all packings is conjectured and still open.

Motivated by their sharp second-level Lasserre hierarchy results for kissing numbers in dimension four, the authors speculate that an appropriate noncompact adaptation may provide a viable approach to resolving the sphere packing problem in dimension four.