Aubin's conjecture on isoperimetric inequalities in nonpositive curvature

Let $(M^m,g)$ be a smooth complete simply-connected Riemannian manifold with sectional curvature at most $\kappa$, where $\kappa\leq 0$. For a bounded domain with smooth boundary in $M$, compare its boundary area with that of a geodesic ball in the simply-connected $m$-dimensional space form $M_\kappa^m$ having the same volume. Aubin's conjecture. The boundary area of the domain is not less than the boundary area of the corresponding geodesic ball in $M_\kappa^m$. The conjecture is known for $\kappa=0$ in dimensions $m\leq 4$ and for $\kappa=-1$ in dimensions $m\leq 3$, but remains open in general.