Lu–Wenzel eigenvalue conjecture

Let $T_X:M(n,\mathbb{C})\to M(n,\mathbb{C})$ be defined by $T_X(Y)=[X^*,[X,Y]]$, and let $\lambda_1(T_X)\geq\cdots\geq\lambda_N(T_X)$ be its eigenvalues. Lu–Wenzel eigenvalue conjecture. For $X\in M(n,\mathbb{K})$ with $\|X\|=1$,

$$\lambda_1(T_X)+\lambda_3(T_X)\leq 3.$$

The supplied text identifies this as a Lu–Wenzel conjecture and says it implies a bound for the case $k=2$ of the related eigenvalue question; no resolution is stated.