Approximating $W^{2,2}$ isometric immersions

Let $S\subset R2$ be a bounded Lipschitz domain and set $$ W^{2,2}_{iso}(S; R3) = \{u\in W^{2,2}(S; R3): (\nabla u)^T(\nabla u) = Id\ \text{ a.e.}\}. $$ Under an additional regularity condition on the boundary $\partial S$ (which is satisfied if it is piecewise continuously differentiable) we prove that the $W^{2,2}$ closure of $W^{2,2}_{iso}(S; R3)\cap C^{\infty}(\bar{S}; R3)$ agrees with $W^{2,2}_{iso}(S; R3)$.