The high-order contact conjecture for stationary varifolds

Let $V$ and $x_0$ satisfy the assumptions of the flat tangent-cone regularity conjecture. The high-order contact conjecture. There is a smooth classical minimal $m$-dimensional graph $\mathcal{M}$ in some neighborhood of $x_0$ such that

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

The conjecture is proposed as a possible route from flat tangent-cone regularity to a unique-continuation problem; the paper does not report a resolution.