The Klee–Wagon conjecture on intersections of congruent balls

Let ${\bf p}=(p_1,\ldots,p_N)$ and ${\bf q}=(q_1,\ldots,q_N)$ be configurations of $N$ points in ${\mathbb E}^d$. The configuration ${\bf q}$ is a contraction of ${\bf p}$ if $|q_i-q_j|\leq |p_i-p_j|$ for every $1\leq i<j\leq N$. Let ${\mathbf B}[p]$ denote the closed unit ball centered at $p$. Klee–Wagon conjecture. If ${\bf q}$ is a contraction of ${\bf p}$, then

$${\rm V}_{d}\left(\bigcap_{i=1}^{N}{\mathbf B}[p_i]\right)\leq {\rm V}_{d}\left(\bigcap_{i=1}^{N}{\mathbf B}[q_i]\right).$$

Thus, contracting the centers should not decrease the volume of the intersection. The paper notes that the conjecture is also considered for non-congruent balls; its general status is open.