Intermediate extension to near-complete thickness

Determine whether, for each fixed dimension d, there exists ε > 0 such that every hypercuboid in LHC_d(n, εn + o(n)) can be extended to a hypercuboid in LHC_d(n, (1 − ε)n + o(n)).

Background

Extendibility concerns adding one layer, whereas completability concerns reaching a full Latin hypercube. The paper asks whether a sparse hypercuboid can instead be extended directly to one whose thickness is close to n, thereby studying an intermediate regime between the two notions.

References

Question 10. For fixed d does there exist a smallε > 0 such that every hypercuboid in LHCd (n,εn + o(n)) can be extended to a hypercuboid in LHCd (n, (1 −ε)n +o(n))?

— Extendibility of Latin Hypercuboids  (2502.08868 - Bowtell et al., 13 Feb 2025) in Question 10, Section 5.2, p. 9