Equitable list point arboricity of graphs

A graph $G$ is list point $k$-arborable if, whenever we are given a $k$-list assignment $L(v)$ of colors for each vertex $v\in V(G)$, we can choose a color $c(v)\in L(v)$ for each vertex $v$ so that each color class induces an acyclic subgraph of $G$, and is equitable list point $k$-arborable if $G$ is list point $k$-arborable and each color appears on at most $\lceil \mathrm{\mid }V(G)\mathrm{\mid }\mathrm{/}k\rceil$ vertices of $G$. In this paper, we conjecture that every graph $G$ is equitable list point $k$-arborable for every $k\ge \lceil (\mathrm{\Delta }(G)+1)\mathrm{/}2\rceil$ and settle this for complete graphs, 2-degenerate graphs, 3-degenerate claw-free graphs with maximum degree at least 4, and planar graphs with maximum degree at least 8.