The bounded-order list vertex arboricity conjecture

Let $G$ be a graph, let $|V(G)|$ denote its number of vertices, and let $\chi_{\mathcal F}(G)$ and $ch_{\mathcal F}(G)$ denote its vertex arboricity and list vertex arboricity, respectively, where $\mathcal F$ is the family of forests.

Bounded-order list vertex arboricity conjecture. If

$$|V(G)|\leq 3\chi_{\mathcal F}(G),$$

then

$$ch_{\mathcal F}(G)=\chi_{\mathcal F}(G).$$

This conjecture was posed in the cited literature and concerns when list vertex arboricity agrees with ordinary vertex arboricity. The source explicitly states that it remains open.