Identical lists maximize the probability of a non-distinguishing coloring

Let $G$ be a graph, and let $\mathcal{L}=\mathcal{L}$ be a collection of equal-sized lists, each of size $k$, indexed by the vertices $v eq 0$ of $G$. Choose a color from each vertex's list uniformly and independently, and let $p(\mathcal{L})$ be the probability that the resulting coloring admits a non-trivial automorphism preserving every color class. List-probability strengthening. The probability $p(\mathcal{L})$ is maximized when all the lists are identical. This would strengthen the List Distinguishing Conjecture by identifying the worst-case list assignment for random colorings.