  We have seen that if \phi: M_n(\C) \rightarrow M_n(\C) is a unital q-positive
map and \nu is a type II Powers weight, then the boundary weight double (\phi,
\nu) induces a unique (up to conjugacy) type II_0 E_0-semigroup. Let \phi:
M_n(\C) \rightarrow M_n(\C) and \psi: M_{n'}(\C) \rightarrow M_{n'}(\C) be
unital rank one q-positive maps, so for some states \rho \in M_n(\C)^* and
\rho' \in M_{n'}(\C)^*, we have \phi(A)=\rho(A)I_n and \psi(D) = \rho'(D)I_{n'}
for all A \in M_n(\C) and D \in M_{n'}(\C). We find that if \nu and \eta are
arbitrary type II Powers weights, then (\phi, \nu) and (\psi, \eta) induce
non-cocycle conjugate E_0-semigroups if \rho and \rho' have different
eigenvalue lists. We then completely classify the q-corners and hyper maximal
q-corners from \phi to \psi, obtaining the following result: If \nu is a type
II Powers weight of the form \nu(\sqrt{I - \Lambda(1)} B \sqrt{I -
\Lambda(1)})=(f,Bf), then the E_0-semigroups induced by (\phi,\nu) and (\psi,
\nu) are cocycle conjugate if and only if n=n' and \phi and \psi are conjugate.
