The commutator function-approximation conjecture for operator monotone functions

Let $\operatorname{CL}_{\lVert\!\lVert\!\lVert\cdot\rVert\!\rVert\!\rVert}([0,\infty))$ denote the class of commutator-Lipschitz functions for the chosen unitarily invariant norm, and define

$$\widetilde E_f(c)=\inf_{g\in\operatorname{CL}_{\lVert\!\lVert\!\lVert\cdot\rVert\!\rVert\!\rVert}([0,\infty))}\left(\sup_{x\geq0}(f-g)(x)-\inf_{x\geq0}(f-g)(x)+c\lVert\!\lVert\!\lVert g\rVert\!\rVert\!\rVert_{\operatorname{CL}([0,\infty))}\right).$$

Let $f$ be a non-negative function that is operator monotone on $[0,\infty)$. The commutator function-approximation conjecture. For every $c>0$,

$$\widetilde E_f(c)\leq Cf(c)$$

with $C=1$. This conjecture is presented as a sufficient reduction for the desired commutator inequality; its resolution is not stated in the supplied excerpt.