Polynomial bounds on maximum degree and maximum face size for excluded minors on a surface

Determine whether there exists a polynomial function Δ_P(g) such that, for every surface S' of Euler genus g, every minimal excluded minor G for S' and the embedding Π considered in the paper, both the maximum vertex degree Δ(G) and the maximum face size Δ_F(G, Π) are bounded by Δ_P(g).

Background

The paper proves quasi-polynomial upper bounds g^{O(log^2 g)} for both the maximum degree of G and the maximum face size in the relevant embedding (G, Π). These parameters are key bottlenecks in pushing the overall size bound from quasi-polynomial to polynomial.

The authors explicitly pose as an open problem whether these structural parameters can be bounded polynomially in the Euler genus, which would be a necessary step toward a polynomial bound on the order of excluded minors.