Morita equivalence of continuous-trace subquotients of exponential solvable Lie group $C^*$-algebras

Let $G$) be an exponential solvable Lie group, and let ${\mathcal J}_1\subseteq{\mathcal J}_2$ be closed two-sided ideals of $C^*(G)$ such that ${\mathcal J}_2/{\mathcal J}_1$ is a $C^*$-algebra with continuous trace. Subquotient Morita-equivalence conjecture. The quotient ${\mathcal J}_2/{\mathcal J}_1$ is Morita-equivalent to a commutative $C^*$-algebra.

This conjecture concerns the fine structure of group $C^*$-algebras through their continuous-trace subquotients. Progress is known for recovering nilpotent Lie groups from their corresponding $C^*$-algebras, but the general problem remains open.