Lu–Wenzel SOS conjecture for real Toeplitz matrices

Let $X$ and $Y$ be arbitrary real Toeplitz matrices, let $\langle X,Y\rangle$ denote their Frobenius inner product, and let $\|\cdot\|$ denote the Frobenius norm. Lu–Wenzel conjecture. The form

$$2\|X\|^2\|Y\|^2-2|\langle X,Y\rangle|^2-\|[X,Y]\|^2$$

is a sum of squares of polynomials. This is an SOS question related to generalized Hilbert's seventeenth problem; the supplied text gives no resolution status.