Exact extremal density when the limiting planar Turán density is maximal

Determine whether every planar graph G satisfying Π(G)=3 also satisfies exP(n,G)=3n−6 for all sufficiently large n.

Background

For a planar graph G, the paper defines Π(G)=lim_{n→∞} exP(n,G)/n and notes that 1≤Π(G)≤3. The paper establishes that the limiting density for the forbidden graph 2C_k approaches 3 as k tends to infinity, but it does not determine whether attaining the maximal possible value Π(G)=3 forces the exact extremal planar Turán number to equal the triangulation bound 3n−6 for all sufficiently large orders.

This problem asks whether asymptotic edge density alone determines eventual exact extremality. A positive answer would connect the limiting planar Turán density with the precise large-n extremal formula.