On a latin square problem of fuchs

Fuchs asked: For which integer partitions g₁ +· · · g_n = N does there exist a latin square of side N having n subsquares of sides g₁, …,g_n having no rows, columns, or symbols in common? Only when at most two distinct integer parts are used is the answer known completely; in general even the necessary conditions are elusive. Two conjectures giving plausible sufficient conditions are advanced. The first asserts than whenever the largest three integer parts are the same, such a latin square always exists. The second asserts that whenever the largest part is no larger than n − 2 times the smallest, such a latin square always exists. Partial results on both are established.