Latin squares with five disjoint subsquares

Given an integer partition (h1⁢h2…hk) of n, is it possible to find an order n latin square with k pairwise disjoint subsquares of orders h1,…,hk? This question was posed by Fuchs and has been answered for all partitions with k≤4. In this paper, we answer the question in the case k=5 and expand on results for special cases of this, such as when the largest part is at most three times the smallest part.