Let $S \subset \mathbb{R}^n$ be the boundary of a centrally symmetric convex body and provide $S$ with the geodesic metric given by its embedding in Euclidean space (i.e., the distance between two points is the infinimum of the Euclidean lengths of all rectifiable curves on $S$ that join them).

Question. Is the diameter of $S$ realized by a pair of antipodal points?

I am curious about this question when the metric is the induced metric from euclidean space, but I'm mostly inetrested in the case of a polytope provided the graph metric on the 1-skeleton. In the discrete case I do not care too much about constants.