Almost linear bound 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 linear upper bound in g.

Background

After establishing a polynomial bound O(g^{8+ε}), the authors ask whether a much tighter, nearly linear dependence on the genus could hold for the maximal order of excluded minors.

This question aims to identify whether the optimal growth is close to the linear lower bound Ω(g).