The normalized commutator inequality with optimal constant for matrix monotone functions

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

$$\lVert\!\lVert\!\lVert f(A)X-Xf(B)\rVert\!\rVert\!\rVert\leq C\lVert\!\lVert\!\lVert X\rVert\!\rVert\!\rVert f\left(\frac{\lVert\!\lVert\!\lVert AX-XB\rVert\!\rVert\!\rVert}{\lVert\!\lVert\!\lVert X\rVert\!\rVert\!\rVert}\right)$$

holds with $C=1$. If the unitarily invariant norm is the operator norm, $X$ need not be compact. This is a normalized form of the preceding commutator conjecture and is presented as a related main focus of the paper; the constant-one claim is not resolved in the supplied context.