Nice question! A comment about terminology. The object you are considering is a "balanced frequency square". Frequency squares generalise Latin squares in the sense that each row and column is a permutation of the same multiset of symbols. The term "balanced" means that each symbol occurs the same number of times.
I think these might be counterexamples:
112323
113232
232311
323211
231123
321132
or
11223344
12142433
21241433
24433112
32131244
34412321
43314212
43324121
-Ian