Optimal reverse isodiametric inequality for planar bisections

Let $K\in\mathcal K^2$ be a 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.

Optimal reverse isodiametric inequality.

$$\frac{\mathrm A(K)}{\mathrm D_B(K)^2}\geq\frac{4}{3\sqrt{3}},$$

\nwith equality if and only if $K$ is the isosceles triangle whose different angle equals $\arccos(\sqrt{2/3})$.

This conjecture proposes the optimal reverse isodiametric-type bound for bisections, motivated by the characterization of the unique triangle in Behrend-bisecting position and the known results on the isodiametric quotient of triangles. The equality case is expected to identify the extremal planar body.