Euclidean distance cone faces

The question remains open whether all faces of the cone of Euclidean distance matrices $LaTeX: \,\mathbb{EDM}^N\!$

(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 $LaTeX: \,\mathbb{EDM}^N\!$ can be defined by intersection with a supporting hyperplane; that intersection is termed exposure.