The radial-density hypersphere conjecture in Euclidean space

Let $\mathbb{R}^{n}$ carry radial density $r^{p}$, with $p>0$, where $r$ is the distance from the origin. A hypersphere through the origin is a sphere whose boundary contains the origin.

Radial-density hypersphere conjecture. In $\mathbb{R}^{n}$ with density $r^{p}$, hyperspheres through the origin are uniquely isoperimetric.

This is presented as the higher-dimensional analogue of the planar problem solved by Dahlberg and collaborators. The supplied text does not state whether the conjecture has been resolved.