Let $K$ and $L$ be planar convex bodies which are not ellipses. Does there exist an affine image $K'$ of $K$ such that

1. $K' \subset L$
2. No ellipse $E$ satisfies $K' \subset E \subset L$

I am also interested in the variant where $K$ and $L$ are centrally symmetric, and $K'$ is required to be a linear image of $K$.