Subalgebras of finite minimal majority algebras revisited

If A is a minimal algebra then the set S_2(A) of all subalgebras of A\times A has a natural structure of ordered involutive monoid. In recent years I have tried to find an abstract characterization of S_2(A) provided A is a finite majority algebra. For some time I have been quite sure that this characterization consists of two conditions: distributivity and completeness. Recently it turned out that one more condition has to be added.