number of overlaps in sphere covering

Question

I have a problem and I'm hoping someone can point me in the right direction. I have the following conjecture:

For a set of spheres of arbitrary radii in $\mathbb{R}^d$, in which the center point of every sphere lies outside of every other sphere, there exists no region in the space that is overlapped by more than $2d$ spheres.

Is this right? Anyone have any thoughts on if a proof for this exists?

Thanks.

Answer 1

Counterexample for $d=2$: You can place five open unit disks on the vertices of a pentagon of side length one. They overlap around the origin.