The radial-measure simplex conjecture

Let $\mu$ be a radially symmetric measure on the ambient Euclidean space with positive density $\rho(r)$ that is monotone decreasing in the radius. Let $B_0(r)$ be the ball of radius $r$ centered at the origin, and let $S$ range over simplices containing this ball. The radial-measure simplex conjecture. The value $\mu(S)$ is minimized at the regular simplex with inscribed ball $B_0(r)$. This is proposed as a stronger version of the Gaussian simplex measure conjecture and is left open in the paper.