Lu–Wenzel fundamental conjecture for real matrices

Let $M(n,\mathbb{R})$ be the space of real $n\times n$ matrices. Let $B,B_2,\ldots,B_m\in M(n,\mathbb{R})$, with $B_\alpha^*$ denoting transpose and with the Frobenius inner product $\operatorname{Tr}(B_\alpha B_\beta^*)$. Assume

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

for every distinct pair among the matrices, and assume

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

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

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

This is the central real Lu–Wenzel conjecture and is intended to unify the BW and DDVV inequalities. The source abstract states that this conjecture is proved in the paper, so its status is solved.