Projectional exposedness of hyperbolicity cones

Determine whether every hyperbolicity cone is projectionally exposed.

Background

Homogeneous cones are hyperbolicity cones, but hyperbolicity cones form a strictly larger class. The paper notes that the strongest general result currently available for hyperbolicity cones is amenability, which is weaker than projectional exposedness.

The unresolved question is whether projectional exposedness extends from homogeneous cones to all hyperbolicity cones.

References

However, the strongest result so far is that hyperbolicity cones are amenable Theorem~6 and it is not known whether they are p-exposed in general.

— Faces of homogeneous cones and applications to homogeneous chordality  (2501.09581 - Gouveia et al., 16 Jan 2025) in Section 1, Introduction; reiterated in Section 4, Conclusion and open questions, item (c)