Polynomial eta-boundedness of $P_5$-free graphs

Let $P_5$ be the path on five vertices. A graph is $P_5$-free if it has no induced subgraph isomorphic to $P_5$; write $\omega(G)$ for its clique number and $\eta(G)$ for the minimum size of a set meeting every maximum stable set.

Polynomial $P_5$-free eta-boundedness conjecture. There exists $d\in\mathbb{N}$ such that every $P_5$-free graph $G$ satisfies $\eta(G)\leq\omega(G)^d$.

The source states that this remains open; proving it would imply the Erdős–Hajnal conjecture for $P_5$-free graphs. The paper proves only a super-exponential bound in this setting.