Maximal space-filling efficiency of the diamond 2H hemoglycin rod lattice

Prove or refute that, among all three-dimensional structures, the hemoglycin rod lattice with diamond 2H symmetry encloses the maximal volume of three-dimensional Euclidean space per rod of material, i.e., achieves the most efficient space covering per rod.

Background

The paper analyzes the hemoglycin lattice, modeled as rods arranged with diamond 2H symmetry, and shows that constructing regular truncated tetrahedra (TTAs) around lattice vertices yields a near-complete space covering with efficiency 23/24 (95.8333%). While this demonstrates exceptional space-filling efficiency, it does not constitute a proof of global optimality.

The authors explicitly state that they cannot prove the original proposition that the diamond 2H rod lattice achieves the maximum enclosed volume per rod among all structures, leaving open a geometric optimization question central to their selection hypothesis for hemoglycin in astrophysical environments.