$\mathbb{Z}_k$-smooth approximation of mod $k$ integral cycles

We prove that every mod $k$ integral cycle $T$ of sufficiently high codimension in an oriented Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by a cycle which is a $\mathbb{Z}_k$-smooth submanifold $Σ$ of nearly the same area, up to a singular set of codimension 5. If in addition