Polynomial-time Maximum Independent Set in graphs with bounded induced cycle packing

Let $k$ be a positive integer, and let an $\mathcal{O}_k$-free graph be a graph containing no induced subgraph from the family $\mathcal{O}_k$.

Polynomial-time independent-set conjecture. $\textsc{Maximum Independent Set}$ is solvable in polynomial time in $\mathcal{O}_k$-free graphs.

The conjecture would extend the known polynomial-time result for graphs of bounded odd cycle packing. The paper proves it in the sparse case and gives a quasi-polynomial-time algorithm in general, but does not establish polynomial-time solvability for arbitrary $\mathcal{O}_k$-free graphs.