Universality for transversal powers of Hamilton cycles

Let $k\ge 2$ and let $\mathbf{G}=\{{\mathbf{G}}_{\mathbf{1}},\dots ,{\mathbf{G}}_{\mathbf{m}}\}$ be a collection of graphs on a common vertex set of cardinality $n$. We show that if each graph in $\mathbf{G}$ has minimum degree at least $(1-\frac{1}{2k}+o(1))n$, then for every edge-colouring $\chi$ of the $k$th power of a Hamilton cycle $C_n^k$ with $m$ colours, there is a copy of $C_n^k$ in $\mathbf{G}$ such that $e\in G_{\chi (e)}$ for every edge $e$ in $C_n^k$. This generalises a result of Bowtell, Morris, Pehova, and Staden, who provided asymptotically best possible minimum degree conditions for the Hamilton cycle.