Kempe equivalent list colorings revisited

A Kempe chain on colors $a$ and $b$ is a component of the subgraph induced by colors $a$ and $b$. A Kempe change is the operation of interchanging the colors of some Kempe chains. For a list-assignment $L$ and an $L$-coloring $\phi$, a Kempe change is $L$-valid for $\phi$ if performing the Kempe change yields another $L$-coloring. Two $L$-colorings are $L$-equivalent if we can form one from the other by a sequence of $L$-valid Kempe changes. A degree-assignment is a list-assignment $L$ such that $L\left(v\right)\ge d\left(v\right)$ for every $v\in V\left(G\right)$. Cranston and Mahmoud 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.