Suppose $K$ is a centrally symmetric convex body in $\mathbb{R}^n$ and $E$ is the John's ellipsoid, the ellipsoid of maximal volume inside $K$.

If $E$ and $K$ have exactly $2n$ contact points, say $(\pm x_i)_{i=1}^{n}$, do $(x_i)_{i=1}^n$ form an orthonormal basis for the Euclidean norm indiuced by $E$?

Naively, this statement seems true in two dimensions, but I don't know how to prove it. Or my intuition could be wrong.

Edit: removed another question (whether all points on $\partial K$ extreme points implies exactly $2n$ contact points) as it has an easy negative answer (in comments). Hopefully the remaining question is not as trivial.