Brakke's radial log-convex density conjecture

Let $\mathbb{R}^{n}$ be equipped with a smooth, radial, log-convex density, and consider regions of prescribed volume. Brakke's conjecture. Balls centered at the origin are isoperimetric regions for every prescribed volume. This is an isoperimetric conjecture for radial log-convex densities and is presented in the source as motivating further study of weighted Faber–Krahn inequalities; the supplied text does not state whether it has been resolved.