Linear extendibility lower bounds in higher dimensions

Determine, for every dimension d > 4, whether there exists a constant γ_d > 0 such that NE_d(n) ≥ γ_d n for all n > 5.

Background

Beyond dimension four, the paper states that essentially no corresponding results are known. A positive answer would provide a uniform linear lower bound on the thickness required to force the existence of nonextendible Latin hypercuboids in each fixed higher dimension.