Strict separation of extendibility and completability thresholds

Characterize all pairs (d,n) for which NE_d(n) > NC_d(n).

Background

Every completable hypercuboid is extendible, so NC_d(n) ≤ NE_d(n). The paper asks precisely when this inequality is strict, which would identify orders and dimensions where extendibility is strictly easier to obtain than completability.