Regularity from a unique flat tangent cone and density concentration

Let $V$ be a stationary integer rectifiable varifold, let $x_0$ be a point with a unique flat tangent cone, let $\Theta(V,x_0)$ denote its density at $x_0$, and let $\mathbf{B}_r(x_0)$ be the radius-$r$ ball centered at $x_0$. Regularity conjecture. If

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

then $x_0$ is a regular point. This is presented as a sufficient condition that would imply the paper's broader measure-zero conjecture; its resolution is not specified in the provided text.