Incidence correspondence between ball intersections and ball-hull boundaries

Let $K=\{p_1,p_2,\dots,p_n\}$ be a finite set in a normed plane $\mathbb{M}^2$, and let $\lambda\geq\lambda_K$. The ball intersection $\operatorname{bi}(K,\lambda)$ and the ball hull $\operatorname{bh}(K,\lambda)$ are the corresponding ball-intersection and ball-hull constructions for $K$ at radius $\lambda$; an arc of $\operatorname{bi}(K,\lambda)$ is generated by a vertex of $\operatorname{bh}(K,\lambda)$, and a vertex of $\operatorname{bi}(K,\lambda)$ is the center of an arc on the boundary of $\operatorname{bh}(K,\lambda)$. Ball-hull and ball-intersection correspondence conjecture. Every arc of $\operatorname{bi}(K,\lambda)$ is generated by a vertex of $\operatorname{bh}(K,\lambda)$, and every vertex of $\operatorname{bi}(K,\lambda)$ is the center of an arc belonging to the boundary of $\operatorname{bh}(K,\lambda)$.

The statement describes the expected dual correspondence between arcs and vertices of the ball intersection and ball hull. The supplied text gives no resolution or supporting theorem for the claim, and the notation $\operatorname{bi}(K,\lambda)$ is not defined in the provided context.