An FPRAS for counting list packings

Let $\Delta$ be a maximum-degree bound, let $q\ge 2\Delta$, and consider graphs of maximum degree at most $\Delta$. A $q$-list packing assigns colors from each vertex's list so that adjacent vertices receive disjoint color sets. Counting conjecture. For each $\Delta$ and $q\ge 2\Delta$, there is a fully polynomial randomized approximation scheme (FPRAS) for counting the number of $q$-list packings of graphs of maximum degree $\Delta$. This would give an efficient approximate-counting algorithm throughout the stated range, extending the algorithmic study of list packing; the source does not indicate that the conjecture has been resolved.