Polynomial-size bound (or tightness of quasi-polynomial) for excluded minors on a surface

Determine whether there exists a polynomial function P(g) such that, for every surface S' of Euler genus g, every minimal excluded minor G for S' satisfies |V(G)| ≤ P(g); alternatively, ascertain whether the quasi-polynomial upper bound |V(G)| ≤ g^{O(log^3 g)} established here is asymptotically tight for the order of minimal excluded minors for surfaces.

Background

The paper improves Seymour’s double-exponential upper bound on the order of minimal excluded minors for a surface to a quasi-polynomial bound g^{O(log^3 g)}. Despite this progress, it is not clear whether a polynomial upper bound is achievable or whether the quasi-polynomial bound reflects the true asymptotics.

Immediately after presenting the main results and discussing bottlenecks, the authors explicitly state that the question of a polynomial bound versus the necessity of a quasi-polynomial bound remains open, and they reiterate it as an open problem.