Strongly outer actions of amenable groups on $\mathcal{Z}$-stable $C^{\ast }$-algebras

Let $A$ be a separable, unital, simple, $\mathcal{Z}$-stable, nuclear $C^{\ast }$-algebra, and let $\alpha \text{\hspace{0.17em}⁣}:G\to \mathrm{A}\mathrm{u}\mathrm{t}(A)$ be an action of a countable amenable group $G$. If the trace space $T(A)$ is a Bauer simplex and the action of $G$ on ${\mathrm{\partial }}_{e}T(A)$ has finite orbits and Hausdorff orbit space, we show that $\alpha$ is strongly outer if and only if $\alpha \otimes {\mathrm{i}\mathrm{d}}_{\mathcal{Z}}$ has the weak tracial Rokhlin property. If $G$ is moreover residually finite, then these conditions are also equivalent to $\alpha \otimes {\mathrm{i}\mathrm{d}}_{\mathcal{Z}}$ having finite Rokhlin dimension (in fact, at most 2). When the covering dimension of ${\mathrm{\partial }}_{e}T(A)$ is finite, we prove that $\alpha$ is cocycle conjugate to $\alpha \otimes {\mathrm{i}\mathrm{d}}_{\mathcal{Z}}$. In particular, the equivalences above hold for $\alpha$ in place of $\alpha \otimes {\mathrm{i}\mathrm{d}}_{\mathcal{Z}}$.