Lipschitz-interface conjecture for isoperimetric subsets of convex domains

Let $\Omega\subset\mathbb{R}^n$ be a bounded, convex domain, and let $E\subset\Omega$ be an open isoperimetric subset. Lipschitz-interface conjecture. The interface $\Omega\cap\partial E$ is a Lipschitz graph. This would imply smoothness by regularity theory for constant-mean-curvature hypersurfaces, contrasting with the possible singularities of area-minimizing surfaces such as the Simons cone.