Separating polynomial $χ$-boundedness from $χ$-boundedness

Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function $f\colon\mathbb{N}\to\mathbb{N}$ we construct a $\chi$-bounded hereditary class of graphs $\mathcal{C}$ with the property that for every integer $n\ge 2$ there is a graph in $\mathcal{C}$ with clique number at most $n$ and chromatic number at least $f(n)$. In particular, this proves that there are hereditary classes of graphs that are $\chi$-bounded but not polynomially $\chi$-bounded.