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 σ_ε collapsing to a subRiemannian metric σ as ε → 0. We establish C^(k,α) estimates for this flow, that are uniform as ε → 0 and as a consequence prove long time existence for the sub Riemannian mean curvature flow of the graph.