  For a countable amenable group $G$ and a fixed dimension $m\geq 1$, we
investigate when it is possible to embed a $G$-space $X$ into the
$m$-dimensional cubical shift $([0,1]^m)^G$. We focus our attention on systems
that arise as an extension of an almost finite $G$-action on a totally
disconnected space $Y$, in the sense of Matui and Kerr. We show that if such a
$G$-space $X$ has mean dimension less than $m/2$, then $X$ embeds into the
$(m+1)$-dimensional cubical shift. If the distinguished factor $G$-space $Y$ is
assumed to be a subshift of finite type, then this can be improved to an
embedding into the $m$-dimensional cubical shift. This result ought to be
viewed as the generalization of a theorem by Gutman-Tsukamoto for $G=\mathbb Z$
to actions of all amenable groups, and represents the first result supporting
the Lindenstrauss-Tsukamoto conjecture for actions of groups other than
$G=\mathbb{Z}^k$.
