For any set $X$, let $[X]^2 = \big\{\{x,y\}:x\neq y\in X\big\}$.Let $G=(V,E)$ be a simple, undirected graph with infinite chromatic number. Let $\kappa$ be a cardinal with $0 < \kappa < \chi(G)$.

Suppose ${\cal W}$ is a collection of subsets of $V$ such that for all $W, W'\in {\cal W}$ we have $W\subseteq W'$ or $W'\subseteq W$, and for every $W\in{\cal W}$ there is a $\kappa$-coloring of $(W, E\cap[W]^2)$.

Is there a $\kappa$-coloring of $(\bigcup {\cal W},\; E\cap [\bigcup {\cal W}]^2)$?