Regularity of mean curvature flow of graphs on Lie groups free up to step 2

We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics $\sigma_\e$ collapsing to a subRiemannian metric $\sigma_0$ as $\e\to 0$. We establish $C^{k,\alpha}$ estimates for this flow, that are uniform as $\e\to 0$ and as a consequence prove long time existence for the subRiemannian mean curvature flow of the graph. Our proof extend to the setting of every step two Carnot group (not necessarily free) and can be adapted following our previous work in \cite{CCM3} to the total variation flow.