Folklore coupling-independence conjecture for proper list-colorings

Let $G=(V,E)$ be a graph with maximum degree $\Delta$, and let $(L_v)_{v\in V}$ be color lists. A proper list-coloring assigns to each vertex $v$ a color in $L_v$ so that adjacent vertices receive different colors. The associated uniform distribution is called $O_\delta(1)$-coupling independent when its coupling-independence parameter is bounded by a constant depending only on $\delta$.

Folklore conjecture. Let $\delta>0$ be a constant. If

$$\left\vert L_v\right\vert\geq (1+\delta)\Delta+O(1)$$

for every $v\in V$, then the uniform distribution $\mu$ over all proper list-colorings of $G$ is $O_\delta(1)$-coupling independent.

This conjecture would establish coupling independence at essentially the $(1+\delta)\Delta$ list-size threshold for arbitrary, possibly unbounded-degree graphs, and would remove the stronger color-list requirement used by the paper's main rapid-mixing result. The source gives no resolution of the conjecture.