The Kneser–Poulsen conjecture for intersections and unions of unequal balls

Let $n,N\in\mathbb N$, let $(x_i)_{i=1}^N,(y_i)_{i=1}^N\subset\mathbb R^n$ satisfy

$$\|x_i-x_j\|_2\le \|y_i-y_j\|_2,$$

and let $(r_i)_{i=1}^N\subset\mathbb R^+$.

Kneser–Poulsen conjecture for unequal balls. The simultaneous inequalities

$${\rm Vol}\left(\bigcap_{i=1}^N B(x_i,r_i)\right)\ge {\rm Vol}\left(\bigcap_{i=1}^N B(y_i,r_i)\right)$$

and

$${\rm Vol}\left(\bigcup_{i=1}^N B(x_i,r_i)\right)\le {\rm Vol}\left(\bigcup_{i=1}^N B(y_i,r_i)\right)$$

should hold.

The source presents this as a broader Kneser–Poulsen-type conjecture. It notes that various particular cases have been verified, but does not give a general resolution.