  Recently, Kim and Park have found an infinite family of graphs whose squares
are not chromatic-choosable. Xuding Zhu asked whether there is some $k$ such
that all $k$th power graphs are chromatic-choosable. We answer this question in
the negative: we show that there is a positive constant $c$ such that for any
$k$ there is a family of graphs $G$ with $\chi(G^k)$ unbounded and
$\chi_{\ell}(G^k)\geq c \chi(G^k) \log \chi(G^k)$. We also provide an upper
bound, $\chi_{\ell}(G^k)<\chi(G^k)^3$ for $k>1$.
