The unique-continuation conjecture for stationary varifolds

Let $V$ be a stationary varifold, let $x_0\in{\rm spt}(V)\cap U$, and let $\mathcal{M}$ be a classical smooth minimal $m$-dimensional surface 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 unique-continuation conjecture. Then ${\rm spt}(V)\subseteq\mathcal{M}$ in some neighborhood of $x_0$. The paper states that this is open even when $\mathcal{M}$ is an $m$-dimensional plane.