Piercing independent sets in graphs without large induced matching

Given a graph $G$, denote by $h(G)$ the smallest size of a subset of $V(G)$ which intersects every maximum independent set of $G$. We prove that any graph $G$ without induced matching of size $t$ satisfies $h(G)\le \omega (G{)}^{3t-3+o(1)}$. This resolves a conjecture of Hajebi, Li and Spirkl (Hitting all maximum stable sets in $P_5$-free graphs, JCTB 2024).