Random extendibility threshold

Determine for which constants d and c almost all hypercuboids in LHC_d(n, cn + o(n)) are extendible.

Background

The paper states that no results are known on random Latin hypercuboids. This question asks for the density regime in which a randomly selected hypercuboid is almost surely extendible.

References

Question 11. For which constants d, c is it true that almost all hypercuboids inLHCd (n, cn + o(n)) are extendible?

— Extendibility of Latin Hypercuboids  (2502.08868 - Bowtell et al., 13 Feb 2025) in Question 11, Section 5.3, p. 10