List-coloring transfer meta-conjecture for graph classes

Let $\mathcal{G}$ be a natural graph class, and let $k$ be a positive integer. For a graph $G$ with a $k$-assignment $L$, let $\mathcal{C}_L(G)$ be the graph of proper $L$-colorings, with edges joining colorings that differ at one vertex. List-coloring transfer meta-conjecture. If there is a constant $C_{k,\mathcal{G}}$ such that $\operatorname{diam}\mathcal{C}_{[k]}(G)\le C_{k,\mathcal{G}}|G|$ for every $G\in\mathcal{G}$ with the uniform assignment $[k]=\\{1,\ldots,k\\}$, then there is a constant $C'_{k,\mathcal{G}}$ such that $\operatorname{diam}\mathcal{C}_L(G)\le C'_{k,\mathcal{G}}|G|$ for every $G\in\mathcal{G}$ and every $k$-assignment $L$. The paper specifically proposes this for classes of bounded maximum average degree and for planar graphs of prescribed girth; it remains open in this generality.