Non-injectivity of Bures--Wasserstein barycentres in infinite dimensions

We construct a counterexample to the injectivity conjecture of Masarotto et al (2018). Namely, we construct a class of examples of injective covariance operators on an infinite-dimensional separable Hilbert space for which the Bures–Wasserstein barycentre is highly non injective – it has a kernel of infinite dimension.