Almost quadratic bound (and optimality) on the order of excluded minors

Determine whether the maximum number of vertices of a minimal excluded minor for a surface of Euler genus g admits an almost quadratic upper bound in g, and ascertain whether the optimal bound is almost quadratic.

Background

The authors consider the possibility that the true optimal upper bound might be quadratic up to lower-order factors, which would still significantly sharpen the current polynomial bound.

They explicitly pose whether an almost quadratic bound holds and, more specifically, whether this is the optimal asymptotic behavior.