Osculating ellipsoids

Let $K$ be a given smooth, origin-symmetric, strictly convex body in $n$ dimensional euclidean space. At every point $x$ on the boundary of $K$ there exists an origin-symmetric ellipsoid $E_x$ that touches $x$ of second-order, the osculating ellipsoid at $x$. Denote the family of osculating ellipsoids by $F:=\{E_x:x\in\mbox{boundary of}~K\}$. Moreover, set $G:=\{TE: E\in F~\&~T\in SL(n)\}$. Is it true that there exists $\mathcal{E}\in G$ such that $\mathcal{E}\subseteq K.$

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Source : Link , Question Author : K. P , Answer Author : Community