Injectivity conjecture for Fréchet means of covariance operators

Let $\mathcal H$ be the underlying Hilbert space, and let $\Sigma_1,\dots,\Sigma_n$ be covariance operators on $\mathcal H$. Assume that $\Sigma_1$ is injective, and let $\overline\Sigma$ be their Fréchet mean with respect to the Procrustes metric $\Pi$. Injectivity conjecture for Fréchet means. The Fréchet mean $\overline\Sigma$ is injective. In finite dimensions, an injective covariance among the observations implies injectivity of the Fréchet mean. The paper poses the infinite-dimensional analogue as an open question because injectivity is needed for the relevant logarithm and exponential maps, as well as for the multicoupling problem and geodesic principal component analysis.