Given a sequence $(\pi_n)$ of irreducible representations of a liminal $C^*$-algebra $A$, and a sequence $(b_n)$ of trace class operators with $b_n\in \pi_n(A)$, we investigate necessary conditions and sufficient conditions for the existence of a simultaneous lifting $a\in A$ such that, for each $n$, the trace of $\sigma(a)$ is bounded for irreducible representations $\sigma$ in a neighbourhood of $\pi_n$.