  Ahlswede and Cai proved that if a simple graph has nested solutions (NS)
under the edge-isoperimetric problems, and the lexicographic (lex) order
produces NS for its second cartesian power,then the lex order produces NS for
any finite cartesian power. Under very general assumptions, we prove that if a
graph and its second cartesian power have NS,then so does any finite cartesian
power. Harper asked if this is true without any restriction. We also conjecture
that it is. All graphs studied in the literature for which the lex order is
optimal are regular. This lead Bezrukov and Els\"asser to conjecture that if
the lex order is optimal for the second cartesian power, then the original
graph is regular. A counterexample to this conjecture is provided.
