The weak ideal property in tensor products

Let A and B be C*-algebras such that A or B is exact. We describe the largest ideal in A circle times B which has the weak ideal property. For many C*-algebras A and B as above we characterize when the largest ideal in A circle times B which has the weak ideal property is the tensor product of the largest ideals in A and B which have the weak ideal property (this is not always true if A or B is exact). Assume that the C*-algebras A and B have the weak ideal property (and one of them is exact). We characterize (in an interesting particular case and also in general) when A circle times B has the weak ideal property (these two characterizations are totally different in nature).