$C^\infty$ rectifiability of stationary varifolds

In this paper we prove that, for every integers $m\geq 2$ and $n\geq 1$, the support of any stationary $m$-dimensional integer rectifiable varifold $V$ in an open set $U\subset \mathbb R^{m+n}$ is $C^\infty$ rectifiable, namely it can be covered, up to an $H^m$-null set, with countably many $C^\infty$ $m$-dimensional graphs.