Restricted Sums in Abelian Groups

For two subsets A and B of an abelian group let A+B={a+b} be the set of all elements of the group, representable as a sum of an element from A and an element from B. Furthermore, let A^B be the subset of A+B consisting only of those elements representable as a+b with a and b distinct.

If there exists an element c in A+B with a unique representation as c=a+b, then the cardinality of A+B is at least |A|+|B|-1; this was proved by P. Scherk in 1955. I conjecture that under the same assumption one has a similar lower bound for the cardinality of A^B.

Conjecture. If there exists an element c in A+B with a unique representation as c=a+b, then the cardinality of the restricted sum A^B is at least |A|+|B|-3.

An immediate consequence of this would be a new proof of the Erdos-Heilbronn conjecture, and there are other exiting conclusions.

Notice, that if A=B then the existence of an an element c with a unique representation implies that A^B is a proper subset of A+B, as this representation of c necessarily must be of the form c=2a. Moreover if, in addition to A=B, in the underlying group there are no elements of even order, then an element with a unique representation exists if and only if A^B is proper in A+B.

The method of Scherk is not particularly complicated, but depends heavily on the e-transform and thus seems to be inapplicable to restricted summation.