Random completability threshold

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

Background

This is the completability analogue of the random extendibility problem. It seeks the constants governing the asymptotic probability that a random Latin hypercuboid can be completed to a Latin hypercube.

References

Question 12. For which constants d, c is it true that almost all hypercuboids inLHCd (n, cn + o(n)) are completable?

— Extendibility of Latin Hypercuboids  (2502.08868 - Bowtell et al., 13 Feb 2025) in Question 12, Section 5.3, p. 10