The connected stable rank of the purely infinite simple $C^*$-algebras

Suppose that $\mathcal A$ is a unital purely infinite simple $C^*$-algebra. If the class [1] of the unit 1 in $K_0(\mathcal A)$ has torsion, then $\operatorname{csr}(\mathcal A)=\infty$; if [1] is torsion-free in $K_0(\mathcal A)$, then $\operatorname{csr}(\mathcal A)=2$. If $\mathcal A$ is a non-unital purely infinite simple $C^*$-algebra, then $\operatorname{csr}(\mathcal A)=2$.