The partial coloring extension conjecture

Let $G$ be a countable group, let $A \subseteq G$, let $k > 1$ be an integer, and let $y: A \rightarrow k$. A partial coloring is a function defined on $A$, and $\overline{[y]}$ denotes the closure of its orbit in the corresponding coloring space.

Partial coloring extension conjecture. Then $y$ can be extended to a $k$-coloring if and only if $\overline{[y]} \cap k^G$ consists of aperiodic points.

This conjecture seeks a complete criterion for extending arbitrary partial colorings to $k$-colorings. The text notes that the criterion has been proved for slender and cofinite domains, but not in full generality.