Sharp symmetric-body bound for Hausdorff distance from convexity

Let $K\subset\mathbb{R}^2$ be a symmetric convex body with strictly convex $C^1$ boundary. For nonempty compact sets $A,B\subset\mathbb{R}^2$, let $\mathcal{R}_K(a,b)$ be the family of $K$-right triangles whose two specified side lengths are $a$ and $b$, with the degenerate interval cases and $\mathcal{R}_K(0,0)=\{0\}$ defined as in the source. Let $d^{(K)}$ denote Hausdorff distance from a set to its convex hull, measured in the norm with unit ball $K$, and let $\operatorname{vert}(T)$ denote the vertices of $T$. The symmetric-body conjecture. One has

$$d^{(K)}(A+B)\leq 2\max\left\{d^{(K)}(\operatorname{vert}(T)):T\in\mathcal{R}_K(d^{(K)}(A),d^{(K)}(B))\right\}.$$

Moreover, for every pair of nonnegative real numbers $\alpha$ and $\beta$, there exist nonempty compact sets $A,B\subset\mathbb{R}^2$ with $d^{(K)}(A)=\alpha$ and $d^{(K)}(B)=\beta$ for which equality holds. This is proposed as an improvement of the paper's preceding bound; the supplied source gives no resolution status.