The smooth approximation conjecture for stationary varifolds

Let $V$ be a stationary $m$-dimensional varifold in $U\subset \mathbb R^{m+n}$, and let $x_0\in U$ be a point at which $\operatorname{spt}(V)$ has an approximate tangent and

$$\lim_{r\downarrow 0} \frac{\\|V\\| (\\{\Theta(V,\\,\cdot\\,)\neq \Theta (V, x_0)\\}\cap \mathbf{B}_r (x_0))}{r^m} = 0.$$

Smooth approximation conjecture. There is a smooth classical minimal $m$-dimensional graph $\mathcal{M}$ in some neighborhood of $x_0$ such that

$$\int_{\mathbf{B}_r (x_0)} \operatorname{dist} (x, \mathcal{M})^2 \,\mathrm{d}\\|V\\| (x) = o (r^N) \qquad\text{for every } N\in \mathbb N.$$

The conjecture proposes infinite-order approximation of a stationary varifold by a smooth classical minimal graph at points where the support has an approximate tangent and the density is asymptotically constant. It is intended as a first step toward understanding the size of the singular complement of the regular set; the surrounding discussion notes that it is not known whether this complement is $\mathcal{H}^m$-null and expects Hausdorff dimension $m-1$.