Single commutators in $C^*$-algebras

Let $\mathcal{A}$ be a simple unital $C^*$-algebra with real rank zero, nonempty tracial state space $T(\mathcal{A})$, and strict comparison of projections. We prove that every element $T$ in $\mathcal{A}$ with $τ(T)=0$ for all $τ\in T(\mathcal{A})$ is a single commutator if one of the following con