  The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two
vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$
in $G$ is at most 2. Let $\chi(H)$ and $\chi_{\ell}(H)$ be the chromatic number
and the list chromatic number of $H$, respectively. A graph $H$ is called {\em
chromatic-choosable} if $\chi_{\ell} (H) = \chi(H)$. It is an interesting
problem to find graphs that are chromatic-choosable.
  Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001)
proposed the List Square Coloring Conjecture which states that $G^2$ is
chromatic-choosable for every graph $G$. Recently, Kim and Park showed that the
List Square Coloring Conjecture does not hold in general by finding a family of
graphs whose squares are complete multipartite graphs with partite sets of
unbounded size. It is a well-known fact that the List Total Coloring Conjecture
is true if the List Square Coloring Conjecture holds for special class of
bipartite graphs. On the other hand, the counterexamples to the List Square
Coloring Conjecture are not bipartite graphs. Hence a natural question is
whether $G^2$ is chromatic-choosable or not for every bipartite graph $G$.
  In this paper, we give a bipartite graph $G$ such that $\chi_{\ell} (G^2)
\neq \chi(G^2)$. Moreover, we show that the value $\chi_{\ell}(G^2) -
\chi(G^2)$ can be arbitrarily large.
