Improved bound on diameter directions when the lattice diameter exceeds one

Determine whether the number of lattice diameter directions of a lattice polytope with lattice diameter greater than one admits a bound smaller than the general bound established in the paper.

Background

The paper proves a sharp upper bound of (2d2)\binom{2^d}{2} on the number of lattice diameter directions for lattice d-polytopes with lattice diameter one. The authors report that computational experiments suggest a smaller bound when the lattice diameter is greater than one, but they explicitly identify the validity and form of such an improvement as unresolved.

References

Our computational experiments suggest that an even smaller bound holds when $diam(P)>1$, but this remains an open problem.

— On Lattice Diameter Segments and A Discrete Borsuk Partition Problem  (2508.20009 - Brose et al., 27 Aug 2025) in Paragraph immediately preceding Definition of dir(x,y) in Section 4.1, 'Diameter directions of a lattice polytope'