Generalization of the main gradient-flow theorems to Hadamard manifolds

The paper considers generalized gradient flows and convex optimization on entanglement polytopes in Hadamard manifolds. In particular, the preceding discussion uses unitary matrices $k\in U_n$, Hermitian matrices $A,B\in\mathcal{H}_n$, and doubly stochastic operators on Hermitian matrices, including the representation of $A=\Psi(B)$ as a convex combination of unitary conjugates of $B$. Generalization conjecture. The conclusion of Theorem~ (and subsequent Theorem~) holds for any Hadamard manifold. This would extend the paper's principal gradient-flow and optimization conclusions beyond the specific setting established in those theorems; the supplied text does not state whether this generalization has been proved or remains open.