I am interested in what is known about the convexity of isoperimetric domains in compact Cartan-Hadamard manifolds (Riemannian manifolds that are complete and simply-connected and have non-positive sectional curvature).

In particular, we know that for a compact Riemannian manifold $M$ with smooth boundary, isoperimetric domains exist. That is, for each $V \in (0,vol(M))$ there exists $E\subset M$ such that $Vol(E) = V$ and $Area(\partial E) = \inf\{area(\partial E) \, | \, E\subset M, vol(E)=V\}$. We also know that isoperimetric domains are $C^{1,1}$ (loosely speaking) and the mean curvature is constant almost everywhere (the mean curvature is defined almost everywhere for $C^{1,1}$ surfaces).

It is an open problem to show that Cartan-Hadmard manifolds satisfy a Euclidean isoperimetric inequality, so I'm wondering if these manifolds similarly have the property that isoperimetric domains are convex. So, if $M$ is a compact, convex Cartan-Hadamard manifold, is it known if isoperimetric domains in $M$ are convex?

A less restrictive question would be: if $M$ is a compact, convex Cartan-Hadamard manifold, do isoperimetric domains $E_0 \subset M$ have the property that $$Area(\partial \,\overline{Conv(E_0)}) \leq Area(\partial E_0)$$ where $\partial \,\overline{Conv(E_0)}$ is the boundary of the closure of the convex hull of $E_0$? Note that in dimensions greater than 2, the surface area of the convex hull of a domain may be greater than the surface area of the domain, even in the Euclidean case.