Asymptotic maximal plane-saturation ratio conjecture

For a maximal planar graph $G$ on $n$ vertices, let its plane-saturation ratio be the relevant ratio measuring the size of a largest plane-saturated subgraph relative to $G$. Plane-saturation ratio conjecture. If $n$ tends to infinity, the maximum value of the plane-saturation ratio over the set of maximal planar graphs $G$ on $n$ vertices tends to

$$\frac{1}{2}.$$

The paper establishes a lower bound for this maximum but states that asymptotic optimality of the lower bound is only suspected, so the limiting value remains open.