First, it is straightforward to adapt Haslhofer's proof to higher dimensions, using the Sobolev inequality: For any compactly supported $u$ such that $\|\nabla u\|_2 < \infty$, $$ \|u\|_{\frac{2n}{n-2}} \le C\|\nabla u\|_2. $$ I encourage you to try.
There must be a standard reference for this, but it is proved in Appendix B of my paper:
Convergence of riemannian manifolds with integral bounds on curvature. II Annales scientifiques de l’É.N.S. 4e série, tome 25, no 2 (1992), p. 179-199