Consider a convex drawing of a graph $G$. Let us assume that the graph is "rich enough": there are vertices inside the boundary polygon.

My question is as follows: does there exist a simple path in the graph connecting two boundary points in such a way that it separates the boundary polygon in two polygons of smaller perimeter (or at least not greater)?

Equivalently, do there always exist two points on the boundary such that (one of) the shortest path connecting them is not either of the boundary paths (and, ideally, does not intersect the boundary other than in the endpoints)?

The statement obviously does not hold for a non-convex drawing, but I was not able to produce an easy counterexample in the convex case. (I hope I didn't miss something obvious.)

Thank you.