Finite minimal-arc representation conjecture for ball hulls

Let $K=\{p_1,p_2,\dots,p_n\}$ be a finite set in a normed plane $\mathbb{M}^2$, and let $\lambda\geq\lambda_K$. A minimal-arc representation conjecture asserts that there exist balls $B(x_i,\lambda)$, for $i=1,\dots,k$, each containing $K$ and having a sphere that contains some minimal arc meeting points of $K$, such that

$$\operatorname{bh}(K)=\bigcap_{i=1}^k B(x_i,\lambda).$$

The statement would extend results previously proved when either $\lambda\geq\operatorname{diam}(K)$ or the normed plane is strictly convex, and would provide a finite description of ball hulls in general normed planes. The source presents it as a possible result rather than establishing it here.