Characterize the degree-four all-Hermitian inequality cone

Characterize the extremal rays of the dimension-free cone of homogeneous degree-four trace inequalities that are nonnegative for every Hermitian matrix in every dimension.

Background

The paper studies dimension-free cones of trace inequalities for Hermitian matrices and their dual moment cones. In degree four, the authors establish some valid inequalities and exhibit a degree-four inequality that does not admit a sum-of-squares decomposition, but they do not classify the entire cone.

A complete description of the cone, including its extremal rays, would clarify which dimension-free degree-four trace inequalities are irredundant and how non-sums-of-squares inequalities contribute to the facial structure of the cone.