Determine the optimal asymptotic bound on the order of excluded minors for a surface

Determine the optimal asymptotic upper bound, as a function of the Euler genus g, on the maximum number of vertices of a minimal excluded minor for a surface of genus g.

Background

The paper proves a polynomial upper bound O(g^{8+ε}) on the number of vertices of minimal excluded minors for a surface of Euler genus g, narrowing the previous quasi-polynomial gap while the best-known lower bound remains Ω(g).

Despite this progress, the exact asymptotic growth of the maximal order of such excluded minors is unresolved. The authors explicitly state that identifying the optimal bound is still open and suggest it might be close to linear or quadratic.