The planar convex-body factorization conjecture

Let $K$ and $L$ be centrally symmetric convex bodies in $\mathbf{R}^2$. Assume that for every linear map $T$ such that

$$T(K)\subset L,$$

there is an ellipse $\mathcal{E}\subset\mathbf{R}^2$ such that

$$T(K)\subset\mathcal{E}\subset L.$$

Planar ellipse conjecture. Then $K$ or $L$ is an ellipse.

This is formulated as a concrete geometric question equivalent to the real HFP conjecture after reducing to two-dimensional spaces; its resolution is therefore tied to the classification of real Banach-space pairs with the HFP.