Choosability with weak diameter for bounded layered tree-width

Let $G$ be a graph, and let $w$ be a positive integer. A graph is 3-choosable with weak diameter at most $N$ if every list assignment of lists of size $3$ admits a coloring such that each monochromatic component has weak diameter at most $N$ in $G$.

Layered-tree-width weak-diameter conjecture. For every positive integer $w$, there exists a positive integer $N$ such that every graph with layered tree-width at most $w$ is 3-choosable with weak diameter in $G$ at most $N$.

This is a stronger-looking list-colouring assertion than bounded clustering in the absence of a maximum-degree hypothesis. The supplied text does not report a resolution.