Tightness of the planar face-cover bound

Determine whether the O(t^4) upper bound on the size of a face cover for a 3-connected plane rooted graph without a rooted K_{2,t} minor is tight.

Background

The paper proves that every 3-connected plane rooted graph without a rooted K_{2,t} minor admits a face cover of size O(t^4). The authors also construct windmill graphs showing that every valid upper bound must be at least Ω(t^2). Thus, the known bounds differ by a quadratic factor in their dependence on t, and the precise asymptotic growth of the optimal bound remains unresolved.