Gyárfás–Sumner analogue for eta-boundedness

For finite graphs $G$ and $H$, say that $G$ is $H$-free if it has no induced subgraph isomorphic to $H$. Write $\omega(G)$ for the clique number and $\eta(G)$ for the minimum size of a set meeting every maximum stable set. A graph class is $\eta$-bounded if its members satisfy an upper bound on $\eta$ depending only on clique number.

Gyárfás–Sumner analogue for $\eta$-boundedness. For every forest $H$, there exists a function $f:\mathbb{N}\to\mathbb{N}$ such that every $H$-free graph $G$ satisfies $\eta(G)\leq f(\omega(G))$.

The conjecture is motivated by the Gyárfás–Sumner conjecture and is known for stars and induced subgraphs of $P_4$, but remains open in general.