Lu–Wenzel constrained DDVV conjecture

Let $\mathbb{K}=\mathbb{R}$ and let $B_1,\ldots,B_m\in M(n,\mathbb{K})$, where $M(n,\mathbb{K})$ is the space of $n\times n$ matrices over $\mathbb{K}$. Let $[A,B]=AB-BA$ and let $\|\cdot\|$ denote the Frobenius norm. Assume

$$\operatorname{tr}\bigl(B_\alpha[B_\gamma,B_\beta]\bigr)=0$$

for all $1\leq\alpha,\beta,\gamma\leq m$. Lu–Wenzel conjecture. Then

$$\sum_{\alpha,\beta=1}^{m}\|[B_\alpha,B_\beta]\|^2\leq\left(\sum_{\alpha=1}^{m}\|B_\alpha\|^2\right)^2.$$

The paper presents this as one of the Lu–Wenzel conjectures and notes that its restriction to real symmetric matrices is the DDVV inequality; no resolution is supplied.