Euclidean distance cone faces
The question remains open whether all faces of the cone of Euclidean distance matrices
(whose dimension is less than dimension of the cone)
are exposed like they are for the positive semidefinite cone.
For a better explanation, see section 6.5.3 in Cone of Distance Matrices.
Definition of exposure is in Convex Geometry.
Basically, the question asks whether all faces of can be defined by intersection with a supporting hyperplane; that intersection is termed exposure.