Let $M^n$ be a Cartan--Hadamard manifold and $B \subset M$ a geodesic ball. In Kleiner’s proof of the Cartan--Hadamard conjecture in dimension 3, the estimate $$ \max_{\partial E} H_{\partial E} \ge H_c(P(E)) $$ for $E$ with $C^{1,1}$ boundary is applied to an isoperimetric region $E \subset B$ to prove the differential inequality $$ (I_c^{-1} \circ I_B)'(v) \ge 1, $$ where $I_B(v)$ denotes the isoperimetric profile of $B$, i.e., the infimum of the perimeter of sets in $B$ with volume $v$, and $I_c(v)$ is the isoperimetric profile of the model space $\mathbb{M}^n_c$ with constant sectional curvature c.
In dimensions $n \le 7$, $\partial E \cap B$ is smooth, so the argument works. In higher dimensions, however, $\partial E \cap B$ may have a singular set $S_0$ with positive $\mathcal{H}^{n-8}$-measure, so it's not $C^{1,1}$ everywhere.
Question:
Can one really directly apply the mean curvature estimate to prove the differential inequality $$ (I_c^{-1} \circ I_B)'(v) \ge 1 $$ even if the boundary is not $C^{1,1}$ everywhere, or is some approximation essentially unavoidable?
Reference: Manuel Ritoré, "Isoperimetric Inequalities in Riemannian Manifolds ", the mean curvature estimate is Proposition 8.5, the proof that proposition 8.5 implies the Cartan-Hadamard conjecture is in p.376 to p.377 starting with "Let us see that Conjecture 8.3 is true in any dimension if the analogous of Proposition 8.5 is valid".
