  A \emph{Kempe chain} on colors $a$ and $b$ is a component of the subgraph
induced by colors $a$ and $b$. A \emph{Kempe change} is the operation of
interchanging the colors of some Kempe chain. For a list-assignment $L$ and an
$L$-coloring $\varphi$, a Kempe change is \emph{$L$-valid} for $\varphi$ if
performing the Kempe change yields another $L$-coloring. Two $L$-colorings are
\emph{$L$-equivalent} if we can form one from the other by a sequence of
$L$-valid Kempe changes. A \emph{degree-assignment} is a list-assignment $L$
such that $L(v)\ge d(v)$ for every $v\in V(G)$. Cranston and Mahmoud
(\emph{Combinatorica}, 2023) asked: For which graphs $G$ and degree-assignment
$L$ of $G$ is it true that all the $L$-colorings of $G$ are $L$-equivalent? We
prove that for every 4-connected graph $G$ which is not complete and every
degree-assignment $L$ of $G$, all $L$-colorings of $G$ are $L$-equivalent.
