Ge–Li–Lu–Zhou even-eigenvalue-sum 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. Ge–Li–Lu–Zhou conjecture. For $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$, if $X\in M(n,\mathbb{K})$ and $\|X\|=1$, then

$$\sum_{i=1}^{2k}\lambda_i(T_X)\leq 2k+2,\qquad k=1,\ldots,\left\lfloor\frac{n^2}{2}\right\rfloor.$$

This is presented as an equivalent form of the fundamental Lu–Wenzel conjecture, and the supplied text gives no independent resolution status.