Existence of flat blowups at boundary points of anisotropic minimizing hypercurrents

We consider $m$-dimensional currents in $\mathbb{R}^{m+1}$ that almost minimize an anisotropic energy whose integrand is continuous in the space variable and $C^{2,{\rm Dini}}$ in the normal variable. We prove that if the boundary of such a current is differentiable, then the current admits a flat blowup at every boundary point.