The odd-prime-power torus packing conjecture

Let $k$ be an odd prime power and let $m\ge1$. Let $H$ be an induced subgraph of the Cartesian power $C_k^m$ of the cycle $C_k$, with $|V(H)|$ dividing $k^m$. A perfect induced $H$-packing is a collection of vertex-disjoint induced copies of $H$ covering every vertex of $C_k^n$.

Odd-prime-power torus packing conjecture. There exists $n_0$ such that, for all $n \ge n_0$, $C_k^n$ admits a perfect induced $H$-packing.

The paper proves the analogous assertion for even $k$ and gives counterexamples when $k$ is odd and not a prime power. The odd-prime-power case is posed as the remaining natural case, and no resolution is supplied in the given text.