Let $G$ be a finite, simple, and connected graph. We say a vertex $u$ in $G$ resolves two other vertices $v$ and $w$ in $G$ if the distance from $u$ to v is not equal to the distance from $u$ to $w$. A subset of vertices of $G$ is a resolving set of $G$ if there exists a vertex in the set that resolves each pair of vertices in $G$. The metric dimension of $G$ is the smallest cardinality of a resolving set. Let us denote this by $\Lambda(G)$.
A graph is said to be planar if there exists a drawing of it on the plane such that no two edges intersect except at the vertices. Planar graphs have faces, which are contiguous regions enclosed by edges. We call a planar graph maximal planar if all the faces of the graph are enclosed by three edges.
Conjecture: For maximal planar graphs with $n$ vertices we have $\Lambda(G) \approx \lfloor2n/5\rfloor$
Any help to prove/disprove the conjecture would be appreciated. Thanks.