Bounded-part List Colouring Conjecture for edge-colourable graphs

Let $s$ be an integer. An edge $\lambda$-choosable graph is a graph whose line graph is colourable from every list assignment whose list-size pattern is the partition $\lambda$. The bounded-part edge list-colouring conjecture. For every integer $s$, there is an integer $k(s)$ such that, whenever $k\geq k(s)$, $G$ is edge $k$-colourable, and $\lambda$ is a partition of $k$ in which every part has size at most $s$, then $G$ is edge $\lambda$-choosable. This is proposed as a weaker version of the List Colouring Conjecture; the supplied text gives no resolution evidence.