Non-Hausdorff part of measure zero?

Let $G$ be a locally compact group of type I.
By Theorem 4.4.5 of Dixmier’s book on C*-algebras, there exists a dense open subset $U$ of the unitary dual $widehat G$, which is a Hausdorff space.

  • Can this set $U$ be chosen in a way that it’s complement has Plancherel-measure zero?

  • Can this set $U$ be chosen such that for every $piin U$ and every $fin C_c^infty(G)$ the operator $pi(f)$ is trace-class and the map $pimapstomathrm{tr}(pi(f))$ is continuous?