The log-convex density conjecture on hyperbolic space

Let $\mathbb{H}^n$ be hyperbolic space with a smooth, radial, log-convex density, and let spheres about the origin mean geodesic spheres centered at the origin. The log-convex density conjecture on hyperbolic space. Every sphere about the origin is isoperimetric. This conjecture concerns the isoperimetric problem for log-convex densities in hyperbolic space; the paper proves a related theorem when the perimeter density is $\phi(R)\cdot 2\cosh^2(R/2)$, rather than equal to the volume density, while the equal-density case remains open.