If we pick a vertex $v$ and two consecutive neighbors of it, $u_1$ and $u_2$, are we sure that $(u_i, u_{i+1}) \in E$? Note: by consecutive I mean in a planar embedding; otherwise any two neighbors can be consecutive.
My intuition is that if $(u_1, u_2) \notin E$, you can add an edge by going from $u_1$ to $v$ and then to $u_2$, thus violating the maximality property that we assumed:
This new edge is not going to intersect any edge incident to $v$ -- otherwise $u_1$ and $u_2$ are not consecutive neighbors; but can it intersect with other edges?
The answer is no. A counter example is $K_5-e$, i.e., removing any edge from a clique of five vertices.