  Let $\pi$ be an irreducible unitary representation of a finitely generated
nonabelian free group $\Gamma$; suppose $\pi$ is weakly contained in the
regular representation. In 2001 the first and third authors conjectured that
such a representation must be either odd or monotonous or duplicitous. In 2004
they introduced the class of multiplicative representations: this is a large
class of representations obtained by looking at the action of $\Gamma$ on its
Cayley graph.
  In the second paper of this series we showed that some of the multiplicative
representations were monotonous. Here we show that all the other multiplicative
representations are either odd or duplicitous. The conjecture is therefore
established for multiplicative representations.
