Finite free extreme points beyond simplices

Determine whether any bounded real free spectrahedra other than those whose first level is a simplex have finitely many free extreme points modulo unitary equivalence.

Background

For bounded real free spectrahedra, free extreme points generate the entire matrix convex set, but the number of such points can be difficult to quantify. The paper recalls that when the first level of a free spectrahedron is a simplex, the free extreme points are precisely the classical extreme points at level one, so there are finitely many modulo unitary equivalence.

The unresolved question is whether finite collections of free extreme points can occur for any other bounded real free spectrahedra. The paper establishes obstructions and constructs infinite families in several non-simplicial settings, but does not resolve the general classification problem. An equivalent formulation concerns which bounded real free spectrahedra have a free polar dual that is also a free spectrahedron.