The periodic coloring conjecture

Let $N\geqslant 1$, let $V$ be a finite set of primitive, nonzero, mutually incommensurable vectors in $\mathbb{Z}^2$, let $\Sigma$ be a finite color set, and let $\Omega\subset \Sigma^{|V|}\times (\mathbb{Z}/N\mathbb{Z})^2$. For each $v\in V$, choose a fixed primitive vector $\bar v$ orthogonal to $v$. Let $\operatorname{Coloring}(V,N,\Sigma,\Omega)$ be the set of tuples of functions $C_v\colon\mathbb{Z}\to\Sigma$ satisfying the local condition specified by the source, using a two-dimensional clock $\sigma\colon(\mathbb{Z}/N\mathbb{Z})^2\to(\mathbb{Z}/N\mathbb{Z})^2$.

Periodic coloring conjecture. For any $V$, $N$, $\Sigma$, and $\Omega$ as above, the solution set $\operatorname{Coloring}(V,N,\Sigma,\Omega)$ is not aperiodic.

This conjecture is presented as an equivalent reformulation of the periodic tiling conjecture in virtually-$\mathbb{Z}^2$ spaces. The source gives no resolution status.