The hyper aperiodic extension conjecture for partial colorings

Let $G$ be a countable group, let $A \subseteq G$, let $k > 1$ be an integer, and let $y: A \rightarrow k$. Define $y_* \in (k+1)^G$ by setting $y_*(a) = y(a)$ for $a \in A$ and $y_*(g) = k$ for $g \in G - A$.

Hyper aperiodic extension conjecture. Then $y$ can be extended to a hyper aperiodic point in $k^G$ if and only if $\overline{[y_*]} \cap k^G$ consists of aperiodic points.

This conjecture proposes a complete criterion for extending a partial coloring to a hyper aperiodic coloring, generalizing the preceding extension theorem for cofinite complements. Its status is not resolved in the supplied text.