Hajebi–Li–Spirkl conjecture for graphs without large induced matchings

For a graph $G$, let $h(G)$ be the smallest size of a subset of $V(G)$ intersecting every maximum independent set, and let $\omega(G)$ denote its clique number. An induced matching of size $t$ consists of vertices $\{a_i,b_i\}_{i\in [t]}$ whose only edges among them are $a_ib_i$ for $i\in[t]$. Hajebi–Li–Spirkl conjecture. If $G$ has no induced matching of size $t$, then

$$h(G)\leq \omega(G)^{O_t(1)}.$$

This conjecture asks for a polynomial bound on the size of a set piercing every maximum independent set in graphs excluding a fixed induced matching; the paper proves the stronger bound $h(G)\leq \omega(G)^{3t-3+o(1)}$, thereby resolving it.