Completely positive cones: Are they facially exposed?

Whether the n×n completely positive cone for n≥5 is facially exposed is an open problem raised by Berman et al. in 2015. In this paper, we give a class of non-exposed faces of the 5×5 completely positive cone, which implies that the 5×5 completely positive cone is not facially exposed. By embedding the 5×5 completely positive cone into the n×n completely positive cone for n>5, we further show that the n×n completely positive cone is not facially exposed for n>5.