Vertex-transitive graph power decomposition conjecture

For a finite graph $G$, let $G^n$ be the graph with vertex set $V(G)^n$, in which two vertices are adjacent exactly when they differ in one coordinate and the entries in that coordinate are adjacent in $G$. Let $H$ be an induced subgraph of $G$. Vertex-transitive graph power decomposition conjecture. If $G$ is vertex-transitive and every prime factor of $|H|$ divides $|G|$, then there exists a positive integer $n$ such that $G^n$ can be partitioned into induced copies of $H$. This proposes that the hypercube decomposition phenomenon extends to powers of every finite vertex-transitive graph under the stated prime-divisibility condition; the source supplies no resolution.