Spectrahedral-shadow classification in ranks three and four

Determine whether completely positive cones over symmetric cones and their dual copositive cones are spectrahedral shadows when the associated Euclidean Jordan algebras have rank 3 or rank 4.

Background

The paper classifies when completely positive cones over symmetric cones are facially exposed, using the rank and structural decomposition of the associated Euclidean Jordan algebra. It establishes the corresponding facial-exposedness classification for ranks 3 and 4, and notes that this classification agrees with the classification of spectrahedrality.

The unresolved issue concerns the stronger representation property of being a spectrahedral shadow, meaning a projection of a spectrahedron. The paper identifies the rank-3 and rank-4 cases as remaining open for both the completely positive cones and their dual copositive cones.