When is a partial Latin square uniquely completable, but not its completable product?

Let P and Q   be uniquely completable partial Latin squares. It is an open problem to determine necessary and sufficient conditions so that the completable product P⊗QP⊗Q is also uniquely completable. So far, only a few specific examples of P have been given such that the completable product of P   with itself (P⊗PP⊗P) does not have a unique completion. In this paper, we find a whole class of such partial Latin squares.