  We show that the following properties of the C*-algebras in a class
$\mathcal{P}$ are inherited by simple unital ${\rm C^*}$-algebras in the class
of asymptotically tracially in $\mathcal{P}$: $(1)$ $\beta$-comparison (in the
sense of Kirchberg and R{\o}rdam), $(2)$ $n$-comparison (in the sense of
Winter).
