Let $K \subset \mathbb{R}^d$ be a convex body, symmetric about the origin and with nonempty interior. Then John's theorem asserts that there exists a unique ellipsoid $E$ of minimal volume such that $K \subset E$.

My question: let $K, K'$ be two such convex sets (that is, symmetric with nonempty interior), and apply John's theorem to get the John's ellipsoids $E, E'$ respectively. Suppose that $K \subset K'$. Is it true, then, that $E \subset E'$?

I have been concerned with how John's ellipsoid varies as you vary the convex set $K$, in particular how the gauge function of $E$ varies. Suppose my claim is true: then it would follow that for any two convex bodies $K_1, K_2$ (again, symmetric about the origin and with nonempty interior) with John's ellipsoids $E_1, E_2$, we'd have $$ \sup_{v \in \mathbb{R}^d} \left| 1 - \frac{n_{E_1}(v)}{n_{E_2}(v)} \right| \leq \sup_{v \in \mathbb{R}^d} \left| 1 - \frac{n_{K_1}(v)}{n_{K_2}(v)} \right| $$ where $n_K$ is the gauge function of a convex body $K$.