Spectrahedrality of Eulerian rigidly convex sets

Determine whether the rigidly convex sets associated with multivariate Eulerian polynomials are spectrahedra.

Background

The paper constructs multivariate real-zero Eulerian polynomials and applies a spectrahedral-relaxation construction to their associated rigidly convex sets. Although the relaxation supplies outer approximations, it does not establish that the rigidly convex sets themselves admit exact spectrahedral representations.

This question is foundational for understanding whether the approximations studied in the paper can eventually be replaced by exact semidefinite descriptions of the Eulerian rigidly convex sets.