The Böttcher–Wenzel conjecture for matrix commutators

Let $X$ and $Y$ be two $n\times n$ matrices, let $[X,Y]=XY-YX$ be their commutator, and let $||\cdot||$ denote the Hilbert–Schmidt norm. Böttcher–Wenzel conjecture.

$$||[X,Y]||^2\leq 2||X||^2||Y||^2.$$

The conjecture arose in random matrix theory and is purely linear algebraic. The paper states that it proves the conjecture; an earlier weaker bound with constant $3$ had been proved by Böttcher and Wenzel.