Separating polynomial $\chi$-boundedness from $\chi$-boundedness

Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function $f\text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{N}\cup \{\mathrm{\infty }\}$ with $f(1)=1$ and $f(n)\ge \left(\genfrac{}{}{0px}{}{3n+1}{3}\right)$, we construct a hereditary class of graphs $\mathcal{G}$ such that the maximum chromatic number of a graph in $\mathcal{G}$ with clique number $n$ is equal to $f(n)$ for every $n\in \mathbb{N}$. In particular, we prove that there exist hereditary classes of graphs that are $\chi$-bounded but not polynomially $\chi$-bounded.