In the above computations, the key point in the collapsed cases is to reduce the metric convergence to the convergence of the harmonic function $h$ and the current $\psi$ by passing to the local universal cover. This can be done when we rescale the metric such that the $C^{k,\alpha}$-geometry is uniformly bounded. In fact, this is exactly the reason why we introduce the notion of $C^{k,\alpha}$-regularity scale.