Polynomial-time MIS conjecture for graphs forbidding independent planar minors

Let $H$ be a planar graph and let $k$ be a positive integer. A graph is $kH$-free if it contains no $k$ pairwise non-adjacent independent $H$-minor models. MIS denotes the Maximum Independent Set problem.

Independent-planar-minor MIS conjecture. For every planar $H$ and every $k$, MIS is polynomial-time solvable on the class of $kH$-free graphs.

The paper proves an $n^{O(\log n)}$-time algorithm for this problem, so the conjecture asks whether the quasi-polynomial bound can always be improved to polynomial time. It remains open in the $2K_3$-free case.