A constant-factor bound for the list chromatic packing number

Let $G$ be a graph. For a list-assignment $L$ of $G$, an $L$-packing is a collection of mutually disjoint $L$-colourings, and it is proper if each colouring is proper. Let $\chi^\star_\ell(G)$ be the least $k$ such that every $k$-list-assignment of $G$ admits a proper $L$-packing of size $k$; let $\chi_\ell(G)$ denote the list-chromatic number of $G$. Constant-factor packing conjecture. There exists $C>0$ such that

$$\chi^\star_\ell(G) \le C \cdot\chi_\ell(G)$$

for any graph $G$. This would reduce the study of list chromatic packing to the ordinary list-chromatic number and would make many list-packing analogues of basic list-colouring results unnecessary; the source does not provide evidence resolving the conjecture.