Embedding conjecture for -actions

Let $(X,\mathbb{Z}^k)$ be a dynamical system, and for each subgroup $A$ of $\mathbb{Z}^k$ define

$$X_A=\{x\in X:nx=x,\ n\in A\}.$$

The quotient group $\mathbb{Z}^k/A$ acts naturally on $X_A$; when the quotient is finite, its mean dimension is $\dim(X_A)/\#(\mathbb{Z}^k/A)$. The $D$-cubical shift is $(([0,1]^D)^{\mathbb{Z}^k},\sigma)$. Embedding conjecture for $\mathbb{Z}^k$-actions. If, for every subgroup $A$ of $\mathbb{Z}^k$,

$$\operatorname{mdim}(X_A,\mathbb{Z}^k/A)<\frac{D}{2},$$

then $(X,\mathbb{Z}^k)$ embeds into the $D$-cubical shift $(([0,1]^D)^{\mathbb{Z}^k},\sigma)$. This conjecture is presented as a general embedding conjecture; the supplied text does not establish its resolution.