The Gaussian simplex measure conjecture

Let $B_0(r)$ be the ball of radius $r$ centered at the origin, and let $S$ range over simplices containing this ball. Let the Gaussian measure have density proportional to $e^{-|x|^2}$. The Gaussian simplex measure conjecture. The Gaussian measure of $S$ is minimized at the regular simplex with inscribed ball $B_0(r)$. The claim is known in dimension $3$ by slicing and the result about spherical caps, but remains open in general.