The list-recoloring conjecture for planar graphs

Let $G$ be a planar graph on $n$ vertices, let $L$ be a list assignment for $G$, and let

and $'$ be $L$-colorings of $G$. A proper $L$-coloring assigns to each vertex a color from its list, with adjacent vertices receiving different colors. **List-recoloring conjecture.** If either - $|L(v)|\ge 10$ for every $v\in V(G)$, or - $G$ is triangle-free and $|L(v)|\ge 7$ for every $v\in V(G)$, then

can be transformed to $'$ by $O(n)$ recolorings, with every intermediate coloring a proper $L$-coloring.

The paper notes that its ordinary-coloring theorems likely generalize to list coloring, but does not establish this statement. The claim would extend the linear-diameter recoloring results from fixed color sets to vertex-specific lists.