Homogeneous symmetric trace-positive sextics with vanishing abelianization are sums of Hermitian squares and commutators

Let $\mathbb{R}\langle X,Y\rangle$ be the free algebra on two symmetric letters. A polynomial is trace-positive if it has non-negative trace on every pair of real symmetric matrices, and it lies in the cone $Θ$ if it is cyclically equivalent to a sum of Hermitian squares. Burgdorf and Klep proved th