The measure-zero conjecture for stationary varifolds

Let $V$ be a stationary integer rectifiable $m$-dimensional varifold, and let its interior singular set be the nonregular points in the interior. The measure-zero conjecture. The interior singular set is $\mathcal{H}^m$-null. The paper explicitly says that this more modest conjecture is still open, despite the time elapsed since Allard's interior regularity theory.