General trace commutator inequality for functions

Let $A$ and $B$ be positive semidefinite $d\times d$ matrices, set $a=\operatorname{Tr}A$ and $b=\operatorname{Tr}B$, and let $f:\mathbb{R}\to\mathbb{R}$ be in a class of functions to be determined. Define

$$F(x)=\int_0^x f(y)\,dy.$$

General trace commutator conjecture. There exists a constant $c$, independent of $d$, such that

$$\lVert[B,f(A+B)]\rVert_1\leq c\bigl(F(a+b)-F(a)-F(b)\bigr).$$

The source reports numerical evidence for $c=1$ for $f(x)=\log x$, $x^2$, and $x^p$ with $0<p\leq1$ under $a+b=1$, and explicitly leaves the admissible class of functions to be determined.