Realizability of product-region Gerechte frameworks

Prove that every Gerechte framework consisting of n=∏_{i=1}^d a_i Cartesian-product regions whose side lengths, up to permutation for each region, are b_i=n/a_i is realizable by an (n,d,d−1,1) Latin hypercube in which every region contains each symbol exactly once.

Background

A Gerechte framework partitions a d-dimensional array of order n into n regions, each containing n^{d−1} cells. It is realizable if there is an (n,d,d−1,1) Latin hypercube such that every region contains every symbol exactly once; a realizable framework together with such a hypercube is a Gerechte design.

The conjecture extends the known two-dimensional result for frameworks whose regions are rectangular s×t or t×s shapes. Theorem 2.1 resolves the special case in which all regions arise from one fixed system of coordinate partitions. The unresolved case permits the coordinate subsets defining the Cartesian-product regions to vary from region to region.