The function-approximation conjecture for operator monotone functions

Let $\widehat{\mathcal M}_+$ be the space of Fourier transforms of finite positive Borel measures on $\mathbb R$, and define

$$E_f(c)=\inf_{g'\in\widehat{\mathcal M}_+}\left(\sup_{x\geq0}(f-g)(x)-\inf_{x\geq0}(f-g)(x)+cg'(0)\right).$$

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

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

with $C=1$. This conjecture is introduced as implying the normalized commutator inequality in the case $A=B$; its resolution is not stated in the supplied excerpt.