  This paper argues that the unitary Cuntz semigroup, introduced in [10] and
termed Cu$_1$, contains crucial information regarding the classification of
non-simple C$^*$-algebras. We exhibit two (non-simple) C$^*$-algebras that
agree on their Cuntz semigroups, termed Cu, and their K$_1$-groups and yet
disagree at level of their unitary Cuntz semigroups. In the process, we
establish that the unitary Cuntz semigroup contains rigorously more information
about non-simple C$^*$-algebras than Cu and K$_1$ alone.
