Can You Take Akemann-Weaver's Diamond Away

In 2004 Akemann and Weaver showed that if Diamond holds, there is a C*-algebra with a unique irreducible representation up to spatial equivalence that is not isomorphic to any algebra of compact operators. This answered, under some additional set-theoretic assumptions, an old question due to Naimark. All known counterexamples to Naimark's Problem have been constructed using a modification of the Akemann-Weaver technique and it was not known whether there exists an algebra of this kind in the absence of Diamond. We show that it is relatively consistent with ZFC that there is a counterexample to Naimark's Problem while Diamond fails.