Polynomial eta-boundedness of matching-free graphs

For each $t\in\mathbb{N}$, let $M_t$ be the unique (up to isomorphism) $1$-regular graph on $2t$ vertices, namely a matching of size $t$. A graph is $M_t$-free if it has no induced subgraph isomorphic to $M_t$; a class is polynomially $\eta$-bounded if its members satisfy $\eta(G)\leq\omega(G)^d$ for some fixed positive integer $d$.

Matching-free polynomial eta-boundedness conjecture. For every integer $t\geq 3$, $M_t$-free graphs are polynomially $\eta$-bounded.

The source notes that even eta-boundedness is not known for $M_3$-free graphs, so the conjecture remains open.