Planar central-symmetry conjecture for equichordal supporting chords

Let $K,L\subset\mathbb R^2$ be convex bodies with $L\subset\operatorname{int}K$, and suppose that $L$ is centrally symmetric. Planar central-symmetry conjecture. If every pair of parallel chords of $K$ supporting $L$ has the same length, then $K$ is centrally symmetric. This is the two-dimensional analogue of the preceding characterization theorem, since the corresponding conclusion in dimension at least three is that both bodies are homothetic and concentric ellipsoids. The conjecture is stated as an expected extension to the planar setting.