John ellipsoid half-volume conjecture

Let $K\subset\mathbb{R}^n$ be a convex body, and let $B_K$ denote its John ellipsoid, the maximal-volume ellipsoid contained in $K$. Scaling $B_K$ by a factor about its center means dilating it about the center of $B_K$.

John ellipsoid half-volume conjecture. The ellipsoid obtained by scaling $B_K$ by a factor of $O(\sqrt{n})$ about its center contains at least half of the volume of $K$.

This conjecture asks whether the deterministically constructible John ellipsoid gives the same $O(\sqrt{n})$ bound for the relevant rounding ratio as the inertia ellipsoid. Its status is not established in the supplied source context.