Latin cubes with disjoint subcubes of two orders

Given a partition h1 + h2 +...+ hk = n, a latin square of order n with pairwise disjoint subsquares of orders h1,...,hk is called a realization. When the values hi are of at most two sizes, the existence of a realization has been completely determined. However, the existence of a latin cube with pairwise disjoint subcubes of two orders is only partially solved. In this paper, we determine existence for such latin cubes in almost all cases.