Fundamental Lu–Wenzel conjecture

Let $\mathbb{K}=\mathbb{R}$ and let $B,B_2,\ldots,B_m\in M(n,\mathbb{K})$. Let $B^*$ denote the conjugate transpose, let $\bot$ denote orthogonality for the Frobenius inner product, and let $[A,B]=AB-BA$. Assume that

$$\operatorname{tr}(B_\alpha B_\beta^*)=0$$

for every $\alpha\ne\beta$, and

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

for all $2\leq\alpha,\beta\leq m$. Fundamental conjecture of Lu–Wenzel. Then

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

The paper calls this the fundamental Lu–Wenzel conjecture and states that the BW inequality, the DDVV inequality, and another Lu–Wenzel conjecture follow from it; its resolution status is not specified.