Dyn–Farkhi conjecture on Hausdorff convexification under Minkowski addition

Let $A,B\subset\mathbb{R}^n$ be nonempty compact sets. Define the Minkowski sum by $A+B:=\{a+b:a\in A,b\in B\}$, the convex hull by $\operatorname{co}(A)$, and

$$d(A):=\sup_{x\in\operatorname{co}(A)}\inf_{y\in A}\|x-y\|_2.$$

Dyn–Farkhi conjecture. One has

$$d(A+B)^2\leq d(A)^2+d(B)^2.$$

The conjecture is true for $n=1$ and was proved for $n=2$, but it was disproved for $n\geq 3$ by constructing suitable sets $A$ and $B$.