Separation conjecture for list arborability and equitable list arborability

Let $G$ be a graph and let $k\in\mathbb{N}$. A graph is $k$-list arborable if every $k$-uniform list assignment admits a list colouring whose colour classes induce acyclic graphs. Suppose also that $G$ has a $k$-colouring in which every colour class has cardinality at most

$$\left\lceil \frac{|V(G)|}{k}\right\rceil$$

and induces an acyclic graph.

Separation conjecture. There is a graph $G$ and $k\in\mathbb{N}$ satisfying both properties above, but $G$ is not equitably $k$-list arborable.

The conjecture asserts that ordinary list arborability together with an equitable acyclic colouring need not imply equitable list arborability. The source proposes it as an open question without a resolution.