Asymptotic extendibility threshold in fixed dimensions

Establish, for each dimension d > 3, whether there exists a constant c_d > 0 such that the minimum thickness NEd(n) of a nonextendible d-dimensional Latin hypercuboid satisfies NEd(n) = c_d n + o(n).

Background

NEd(n) denotes the smallest k for which a d-dimensional Latin hypercuboid of order n and thickness k exists that is not extendible. The paper records linear lower bounds in some dimensions and upper bounds in dimension three, but the precise asymptotic growth is unknown in general. This question asks whether a well-defined linear asymptotic constant exists in every dimension above three.