The commutator inequality with optimal constant for matrix monotone functions

Let $A$, $B$, and $X$ be operators in $B({\mathcal H})$. Assume that $A,B\geq 0$ are compact with finite unitarily invariant norm $\lVert\!\lVert\!\lVert\cdot\rVert\!\rVert\!\rVert$ and that $\lVert X\rVert\leq 1$. Let $f$ be a non-negative matrix monotone function on $[0,\infty)$. The commutator inequality. The inequality

$$\lVert\!\lVert\!\lVert f(A)X-Xf(B)\rVert\!\rVert\!\rVert\leq C\lVert\!\lVert\!\lVert f(|AX-XB|)\rVert\!\rVert\!\rVert$$

holds with $C=1$. If the unitarily invariant norm is the operator norm, $A$ and $B$ need not be compact. This conjecture seeks the optimal constant in a commutator analogue of Ando's theorem; the preceding finite-dimensional result gives the currently available bound $C=5/4$, while the constant-one assertion remains unresolved in the stated generality.