Suppose that $\mathcal M$ is a countably decomposable type II$_1$ von Neumann
algebra and $\mathcal A$ is a separable, non-nuclear, unital C$^*$-algebra. We
show that, if $\mathcal M$ has Property $\Gamma$, then the similarity degree of
$\mathcal M$ is less than or equal to $5$. If $\mathcal A$ has Property
c$^*$-$\Gamma$, then the similarity degree of $\mathcal A$ is equal to $3$. In
particular, the similarity degree of a $\mathcal Z$-stable, separable,
non-nuclear, unital C$^*$-algebra is equal to $3$.
