Optimal reverse isodiametric inequality for centrally symmetric planar bisections

Let $K\in\mathcal K^2$ be a centrally symmetric planar convex body in Behrend-bisecting position, meaning that its bisection diameter is minimized by a bisection into two parts that are each in Behrend position.

Centrally symmetric reverse isodiametric inequality.

$$\frac{\mathrm A(K)}{\mathrm D_B(K)^2}\geq 1,$$

\nwith equality if and only if $K=[-1,1]\times[-2,2]$.

The conjecture sharpens the preceding bound for centrally symmetric bodies. The rectangle in the equality case arises from the analysis of centrally symmetric bisections, while the source indicates that the sharpness of the earlier theorem remains unresolved.