Density results relative to the Dirichlet energy of mappings into a manifold

Let Y be a smooth compact oriented Riemannian manifold without boundary. Weak limits of graphs of smooth maps u_k:B^n\to Y with equibounded Dirichlet integral give rise to elements of the space cart^{2,1}(B^n\times Y). Assume that Y is 1-connected and that its 2-homology group has no torsion. In any dimension n we prove that every element T in cart^{2,1}(B^n\times Y) with no singular vertical part can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps u_k:B^n\to Y with Dirichlet energies converging to the energy of T.